All the solutions provided in McGraw Hill My Math Grade 5 Answer Key PDF Chapter 12 Lesson 3 Classify Triangles will give you a clear idea of the concepts.
McGraw-Hill My Math Grade 5 Answer Key Chapter 12 Lesson 3 Classify Triangles
You can classify triangles using one or more of the following attributes. An attribute is a characteristic of a figure like side measures and angle measures.
Math in My World
Example 1
The Hammond family traveled from Columbus, Ohio, to Dallas, Texas, and then to Atlanta, Georgia, before returning home. The distance of each flight is shown on the map. Find the number of congruent sides.
The lengths of the sides of the triangle are 924 miles, 573 miles, and _____________ miles.
How many sides of the triangle are congruent?
So. the triangle formed on the map in Example 1 is a
___________ triangle.
Answer:
The above-given:
The length of the sides are given:
924 miles, 573 miles, and 729 miles.
The number of sides of the triangle is congruent = S
S = 0
There are no congruent sides because no sides are equal.
The triangle formed on the map is a scalene triangle.
Scalene triangle: A triangle in which all three sides are of different lengths and all three angles of the triangle are also of different measures, is called a scalene triangle.
Example 2
Triangles form the sides of the Khafre Pyramid in Egypt. Determine the number of acute, obtuse, or right angles in the triangle.
How many angles of the triangle are acute? _____________
How many angles of the triangle are obtuse? _____________
How many angles of the triangle are right? _____________
So, the triangle in Example 2 is a(n) ______________.
Answer:
First of all, we need to know about the acute, obtuse and right angles.
Acute angle: An angle which measures less than 90° is called an acute angle. The measure is between 0° to 90°.
Obtuse angle: An angle that measures greater than 90° is known as the obtuse angle. The angle measure ranges from 90° to 180°. An obtuse angle can also be found if we have the measure of the acute angle.
Obtuse Angle Measure = (180 – acute angle measure)
Right angle: An angle which measures exactly 90° is called a right angle. It is generally formed when two lines are perpendicular to each other.
Now write the number of angles in the triangle from the above information.
The number of angles of the triangles is acute = 3
The number of angles of the triangle is obtuse = 0
The number of angles of the triangle is right = 0
Finally, the triangle is an acute triangle.
Guided Practice
Question 1.
Classify the triangle based on its sides.
How many sides of the triangle are congruent? ______________
The triangle is a(n) _____________.
Answer:
From the above-given information:
The number of sides of the triangle is congruent = 2
Isosceles triangles are those triangles that have at least two sides of equal measure.
From the above definition, the triangle is isosceles.
Properties:
– Two equal sides and two equal angles.
– The two equal sides of an isosceles triangle are called the legs and the angle between them is called the vertex angle or apex angle.
– The side opposite the vertex angle is called the base and base angles are equal.
– The perpendicular from the apex angle bisects the base and the apex angle.
– The perpendicular drawn from the apex angle divides the isosceles triangle into two congruent triangles and is also known as its line of symmetry.
Question 2.
Classify the triangle based on its angles.
The triangle is a(n) ______________.
Answer:
Classifying the triangle based on its angles: Triangles can also be classified on the basis of angles. All triangles have three interior angles whose angle measurements sum to 180° with different combinations of angles depending on the type of triangle. The three different types are acute triangle, obtuse triangle, and right triangle.
An acute triangle is a type of triangle where all three interior angles are acute angles or less than 90°.
There are no right angles or obtuse angles.
From the above-given information, we can say that the triangle is acute.
Talk Math
Describe an isosceles right triangle.
Answer:
Definition: An Isosceles Right Triangle is a right triangle that consists of two equal-length legs. Since the two legs of the right triangle are equal in length, the corresponding angles would also be congruent. Thus, in an isosceles right triangle, two legs and the two acute angles are congruent.
The formula for Isosceles right triangle:
– Actually, it follows Pythagoras’s theorem to give the relationship between the hypotenuse and the equal sides. Let’s look into the diagram below to understand the isosceles right triangle formula.
In ∆QPR,
PQ = QR = S units [Equal Sides]
PR = l units [Hypotenuse]
Using Pythagoras’ theorem,
Hypotenuse2 = Side2 + Side2
PR2 = PQ2 + QR2
l2 = S2 + S2
l2 = 2S2
Thus, l = S√2 units.
Perimeter: The perimeter of the isosceles right triangle formula is 2x + l.
Properties:
– It has one angle measuring 90º.
– The legs of this triangle are perpendicular to each other which are also known as the base and the height.
– The other two angles of an isosceles right triangle are acute and congruent to each other measuring 45° each.
– The sum of all the interior angles is equal to 180°.
– The altitude drawn from the right angle is the perpendicular bisector of the hypotenuse (opposite side).
– The area of an isosceles right triangle is given as (1/2) × Base × Height square units.
Independent Practice
Determine the number of congruent sides for each triangle. Then classify the triangle based on its sides.
Question 3.
Answer:
From the above-given information:
The number of congruent sides is 0.
Scalene triangle: A triangle in which all three sides are of different lengths and all three angles of the triangle are also of different measures, is called a scalene triangle.
The triangle is right scalene.
Question 4.
Answer:
From the above-given information:
The number of congruent sides = 2
Based on its sides the triangle is isosceles.
An Isosceles triangle is a triangle that has two equal sides. Also, the two angles opposite the two equal sides are equal.
Classify each triangle based on its angles.
Question 5.
Answer:
Classification of triangle based on its angles:
Triangles can also be classified on the basis of angles. All triangles have three interior angles whose angle measurements sum to 180° with different combinations of angles depending on the type of triangle. The three different types are acute triangle, obtuse triangle, and right triangle.
The above-given triangle says that it is an obtuse triangle.
Obtuse triangle: An obtuse-angled triangle or obtuse triangle is a type of triangle whose one of the vertex angles is bigger than 90° and the sum of the other two angles is less than 90°. The side opposite to the obtuse angle is considered the longest.
Question 6.
Answer:
Classification of triangle based on its angles:
Triangles can also be classified on the basis of angles. All triangles have three interior angles whose angle measurements sum to 180° with different combinations of angles depending on the type of triangle. The three different types are acute triangle, obtuse triangle, and right triangle.
The above-given triangle says that it is an acute triangle.
Acute angle: Acute triangle is a type of triangle where all three interior angles are acute angles or less than 90°. The sides of an acute-angled triangle can be equal or unequal depending on whether the triangle is equilateral, isosceles, or scalene.
Question 7.
Answer:
Classification of triangle based on its angles:
Triangles can also be classified on the basis of angles. All triangles have three interior angles whose angle measurements sum to 180° with different combinations of angles depending on the type of triangle. The three different types are acute triangle, obtuse triangle, and right triangle.
The above-given triangle says that it is a right triangle.
Right angle triangle: A right-angled triangle is a triangle with one of the angles at 90 degrees. A 90-degree angle is called a right angle, and hence the triangle with a right angle is called a right triangle.
Question 8.
Answer:
Classification of triangle based on its angles:
Triangles can also be classified on the basis of angles. All triangles have three interior angles whose angle measurements sum to 180° with different combinations of angles depending on the type of triangle. The three different types are acute triangle, obtuse triangle, and right triangle.
The above-given triangle says that it is an acute triangle.
Acute angle: Acute triangle is a type of triangle where all three interior angles are acute angles or less than 90°. The sides of an acute-angled triangle can be equal or unequal depending on whether the triangle is equilateral, isosceles, or scalene.
Draw each triangle.
Question 9.
equilateral triangle
Answer:
The equilateral triangle is shown below:
Definition: an equilateral triangle is a triangle that has all its sides equal in length. Also, the three angles of the equilateral triangle are congruent and equal to 60 degrees. The sum of all three angles of an equilateral triangle is equal to 180 degrees. (60° + 60° + 60° = 180°.)
Properties:
– All three sides are equal.
– All three angles are congruent and are equal to 60 degrees.
– It is a regular polygon with three sides.
– The perpendicular drawn from the vertex of the equilateral triangle to the opposite side bisects it into equal halves. Also, the angle of the vertex from where the perpendicular is drawn is divided into two equal angles, i.e. 30 degrees each.
– The ortho-centre and centroid are at the same point.
– In an equilateral triangle, the median, angle bisector, and altitude for all sides are the same.
– The area of an equilateral triangle is √3a2/ 4
– The perimeter of an equilateral triangle is 3a.
Question 10.
right triangle
Answer:
– A right-angled triangle is a triangle with one of the angles at 90 degrees. A 90-degree angle is called a right angle, and hence the triangle with a right angle is called a right triangle. Here, the relationship between the sides is understood with the help of the Pythagoras rule. The side opposite to the right angle is the largest side and is referred to as the hypotenuse.
– The formula to calculate the area of a right triangle formula is given: Perimeter = a + b + c (where a, b, and c are the three sides of the triangle.)
– The formula to calculate the area of a right triangle formula is given: Area = 1/2 × Base × Height = 1/2 × b × h where height,h is equal to the length of the perpendicular side of the triangle.
– The Pythagoras theorem definition shows the relation among the three sides of a right triangle. The square of the hypotenuse is equal to the sum of the square of the other two sides
(Hypotenuse)2 = (Perpendicular)2 + (Base)2
Problem Solving
Question 11.
Half of a rectangular sandwich looks like a triangle. Classify it based on its angles.
Answer:
From the above-given information:
The triangle has 1 right angle and 2 acute angles.
A right-angled triangle is a triangle with one of the angles at 90 degrees. A 90-degree angle is called a right angle, and hence the triangle with a right angle is called a right triangle.
Therefore, the triangle we can classify based on its angle is a right triangle.
Question 12.
Mathematical PRACTICE Identify Structure Measure the sides of the sandwich. Classify the triangle based on its sides.
Answer:
No sides are congruent so it is a scalene triangle.
Scalene triangle: A triangle in which all three sides are of different lengths and all three angles of the triangle are also of different measures, is called a scalene triangle.
HOT Problems.
Question 13.
Mathematical PRACTICE Draw a Conclusion Emma, Gabriel, Jorge, and Makayla each drew a different triangle. Use the clues below to describe each person’s triangle as isosceles, equilateral, or scalene and also as acute, right, or obtuse.
- Gabriel and Jorge each drew a 900 angle in their triangles.
- Gabriel’s triangle does not have any congruent sides.
- One angle in Emma’s triangle measures greater than 90°.
- Each side of Makayla’s triangle and two sides of Emma’s and Jorge’s triangles are four centimetres long.
Answer:
Simply we can draw a box for better understanding from the above clues:
from the above two diagrams, we can conclude that:
Gabriel has scalene and a right triangle.
Jorge has an Isosceles and a right triangle.
Emma has an Isosceles and an obtuse triangle.
Makayla has an equilateral and acute triangle.
Question 14.
Building on the Essential Question How do I classify triangles using their attributes?
Answer:
Classification of triangles is done according to the length of their sides and the measure of the angles. A triangle is a simple polygon with 3 sides, 3 interior angles, and 3 vertices that are joined with each other and it is denoted by the symbol △.
Classifying the triangles based on their sides:
A triangle consists of three sides but the factor to classify them is the length of the sides. The three types of triangles are equilateral triangle, scalene triangle, and isosceles triangle.
Equilateral triangle: The three sides are equal to each other along with the three angles measuring 60°. An equilateral triangle is considered a regular polygon with angles and sides equal.
Isosceles triangle: Isosceles triangle has 2 equal sides and 2 equal base angles.
Scalene triangle: A scalene triangle is a triangle with all three sides of different lengths along with three angles of different measures.
Classifying the triangle based on its angles:
Triangles can also be classified on the basis of angles. All triangles have three interior angles whose angle measurements sum to 180° with different combinations of angles depending on the type of triangle. The three different types are acute triangle, obtuse triangle, and right triangle.
Acute triangle: Acute triangle is a type of triangle where all three interior angles are acute angles or less than 90°.
Right triangle: A right-angled triangle is a triangle with one of the angles at 90 degrees. A 90-degree angle is called a right angle, and hence the triangle with a right angle is called a right triangle.
Obtuse triangle: An obtuse-angled triangle or obtuse triangle is a type of triangle whose one of the vertex angles is bigger than 90° and the sum of the other two angles is less than 90°.
McGraw Hill My Math Grade 5 Chapter 12 Lesson 3 My Homework Answer Key
Practice
Question 1.
Determine the number of congruent sides. Then classify the triangle based on its sides.
How many sides of the triangle are congruent? ___________
The triangle is a ______________.
Answer:
From the above-given information:
The number of sides of the triangle is congruent = 0
The triangle is a scalene.
Scalene triangle: A scalene triangle is a triangle with all three sides of different lengths along with three angles of different measures.
Vocabulary Check
Fill in each blank with the correct term(s) or number(s) to complete each sentence.
Question 2.
An equilateral triangle is a triangle with ____________ congruent sides.
Answer: 3
Definition of an equilateral triangle: The three sides are equal to each other along with the three angles measuring 60°. An equilateral triangle is considered a regular polygon with angles and sides equal.
Question 3.
An acute triangle is a triangle with angles each less than ______________.
Answer: 90 degrees.
Acute triangle: Acute triangle is a type of triangle where all three interior angles are acute angles or less than 90°.
Question 4.
An obtuse triangle is a triangle with one angle that is greater than ______________.
Answer: 90 degrees.
Obtuse triangle: An obtuse-angled triangle or obtuse triangle is a type of triangle whose one of the vertex angles is bigger than 90° and the sum of the other two angles is less than 90°.
Problem Solving
Question 5.
Look at the triangle on the top of the White House in the photo. Describe the sides and angles of the triangle.
Answer:
The above-given information:
Observe the triangle, it has 2 congruent sides.
Coming to the angles, it has 2 acute angles and 1 obtuse angle.
Question 6.
Serena has an art easel with sides of equal length. She opened the easel and placed it on her desk. Classify the type of triangle formed by the easel and the desk according to its sides. Next, classify the type of triangle formed by the easel and the desk according to its angles.
Answer:
The above-given:
The length is equal on all sides.
she is placed on the desk then the triangle forms:
Here 2 sides are equal so it is an isosceles triangle. And the angles are acute.
* Isosceles triangle: Isosceles triangle has 2 equal sides and 2 equal base angles.
Question 7.
Mathematical PRACTICE Identify Structure The image shown at the right contains many triangles. Describe the different types of triangles found in the image.
Answer:
We can write the different triangles because the picture presents various sides and angles.
By observing the picture, we can say that it has an Isosceles triangle, scalene, acute, right, and obtuse.
we classified based on its sides and angles.
Question 8.
Mathematical PRACTICE Justify Conclusions A triangle has two sides that are perpendicular. Could the triangle be isosceles, equilateral, or scalene? Explain.
Answer:
It could be an Isosceles triangle or scalene triangle. Because it depends on the length.
Coming to the angles it is a right triangle. Because it has 1 right angle and 2 acute angles.
This type of triangle has two sides of the same length. The two equal sides of an isosceles triangle are called the legs and the angle between them is called the vertex angle or apex angle.
Test Practice
Question 9.
Which of the following figures is an obtuse triangle?
Answer:
Obtuse triangle: An obtuse-angled triangle or obtuse triangle is a type of triangle whose one of the vertex angles is bigger than 90° and the sum of the other two angles is less than 90°.