High School Math : 45/45/90 Right Isosceles Triangles

Study concepts, example questions & explanations for High School Math

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Example Questions

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Example Question #1 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem

Isosceles

In an isosceles right triangle, two sides equal . Find the length of side

Possible Answers:

Correct answer:

Explanation:

This problem represents the definition of the side lengths of an isosceles right triangle.  By definition the sides equal , and .  However, if you did not remember this definition one can also find the length of the side using the Pythagorean theorem .

Example Question #2 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem

ABCD is a square whose side is  units.  Find the length of diagonal AC.

Possible Answers:

 

none of the other answers

Correct answer:

Explanation:

To find the length of the diagonal, given two sides of the square, we can create two equal triangles from the square. The diagonal line splits the right angles of the square in half, creating two triangles with the angles of , , and  degrees. This type of triangle is a special right triangle, with the relationship between the side opposite the  degree angles serving as x, and the side opposite the  degree angle serving as .

 

Appyling this, if we plug  in for  we get that the side opposite the right angle (aka the diagonal) is 

Example Question #3 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem

The area of a square is . Find the length of the diagonal of the square.

Possible Answers:

Correct answer:

Explanation:

If the area of the square is , we know that each side of the square is , because the area of a square is .

Then, the diagonal creates two special right triangles. Knowing that the sides = , we can find that the hypotenuse (aka diagonal) is 

Example Question #1 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Isosceles

An isosceles right triangle has a hypotenuse of .  Find its area.

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

In order to calculate the triangle's area, we need to find the lengths of its legs.  An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as  triangles and their side lengths follow a specific pattern that states that one can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2.

Now we can calculate the area using the formula

 

Now, convert to feet.

Example Question #2 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

The base of a right isosceles triangle is 8 inches.  The hypotenuse is not the base.  What is the area of the triangle in inches?

Possible Answers:

Correct answer:

Explanation:

To find the area of a triangle, multiply the base by the height, then divide by 2.  Since the short legs of an isosceles triangle are the same length, we need to know only one to know the other.  Since, a short side serves as the base of the triangle, the other short side tells us the height.

 

Example Question #1 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle

Isosceles

The hypotenuse of an isosceles right triangle has a measure of . Find its perimeter.

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

In order to calculate the triangle's perimeter, we need to find the lengths of its legs.  An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as  triangles and their side lengths follow a specific pattern that states that one can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2.

Now we can calculate the perimeter by doubling  and adding .

Example Question #1 : 45/45/90 Right Isosceles Triangles

Isosceles

The side lengths of an isoceles right triangle measure . Find its perimeter.

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

An isosceles triangle is a special triangle due to the values of its angles.  These triangles are referred to as  triangles and their side lenghts follow a specific pattern that states you can calculate the length of the hypotenuse of an isoceles triangle by multiplying the length of one of the legs by the square root of 2.

Now we can calculate the perimeter by doubling  and adding .

Example Question #3 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle

A triangle has two angles equal to and two sides equal to . What is the perimeter of this triangle?

Possible Answers:

Correct answer:

Explanation:

When a triangle has two angles equal to , it must be a isosceles right triangle.

The pattern for the sides of a is .

Since two sides are equal to , this triangle will have sides of .

Add them all together to get .

Example Question #1 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle

An isosceles triangle has a base of 6 and a height of 4. What is the perimeter of the triangle?

Possible Answers:

None of these

Correct answer:

Explanation:

An isosceles triangle is basically two right triangles stuck together. The isosceles triangle has a base of 6, which means that from the midpoint of the base to one of the angles, the length is 3. Now, you have a right triangle with a base of 3 and a height of 4. The hypotenuse of this right triangle, which is one of the two congruent sides of the isosceles triangle, is 5 units long (according to the Pythagorean Theorem).

The total perimeter will be the length of the base (6) plus the length of the hypotenuse of each right triangle (5).

5 + 5 + 6 = 16

Example Question #1 : Isosceles Triangles

What is the area of a square that has a diagonal whose endpoints in the coordinate plane are located at (-8, 6) and (2, -4)?

Possible Answers:

50√2

100√2

200√2

100

50

Correct answer:

100

Explanation:

Square_part1

Square_part2

Square_part3

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