Right Triangles Quiz | Math

Quiz Answer Key and Fun Facts

1. A right triangle ABC is given where angle B = 90 degrees. Which side of the triangle is the longest?

Answer: side CA

In the given triangle ABC,
angle B = 90 degrees.

In a right triangle, the side opposite the right angle (which in this case is angle B), is always the longest. Hence, AC is the longest side of the triangle.

The longest side of a right triangle is commonly referred to as the ~hypotenuse~.

2. There is a right triangle PQR where: angle Q = 90 degrees; angle P = angle R. What is the measure of angles P and R?

Answer: 45 degrees

In the given triangle PQR,
angle Q = 90 degrees.

Now let us assume:
angle P = angle R = x.

By the angle sum property of a triangle, the sum of the three angles of a triangle is equal to 180 degrees.
=> angle P + angle Q + angle R = 180 degrees
=> x + x + 90 degrees = 180 degrees
=> 2x = 90 degrees
=> x = 45 degrees

Hence, *both angle P and angle R = 45 degrees*.

3. The Pythagorean Theorem states that: "The square of the hypotenuse of a right angle triangle is equal to the sum of the squares of the other two sides." This means that, in a right triangle ABC with hypotenuse AC:

Answer: AB^2 + BC^2 = AC^2

Let me try to explain this with the help of a triangle ABC, where:
AC is the hypotenuse;
AB and BC are the other sides.

The Pythagorean theorem states that: "The square of the hypotenuse of a right angle triangle is equal to the sum of the squares of the other two sides."

>> The square of the hypotenuse...
Here AC is the hypotenuse. By squaring it we get AC^2.

>> ...the sum of the squares of the other two sides.
The other two sides are AB and BC. If we square them we get AB^2 and BC^2. And their sum is: AB^2 + AC^2.

>> ...is equal to...:
With this we can conclude that
*AB^2 + BC^2 = AC^2*.

4. A right triangle LMN is given where: side MN = 4 side NL (the hypotenuse) = 5 What is the length of side LM?

Answer: 3

We have with us a right triangle and a hypotenuse (NL = 5), and one of the other sides (MN = 4). We have to find the length of side LM.

By the Pythagorean Theorem,

LM^2 + MN^2 = NL^2
=> LM^2 + 4^2 = 5^2
=> LM^2 + 16 = 25
=> LM^2 = 9
=> *LM = 3*

5. Two right triangles are congruent if the hypotenuse and one side of triangle are congruent to the hypotenuse and the corresponding side of the other triangle respectively.

Answer: True

This is commonly known as the RHS (Right-Hypotenuse-Side) criterion for congruence of right triangles.

Two right triangles ABC and PQR will be congruent by RHS criterion if:
angle B = angle Q = 90 degrees (right angles);
side AC = side PR; (hypotenuses); and
side AB = side PQ (or side BC = side QR).

6. Out of the following triangles, which is *NOT* a right triangle?

Answer: Triangle PQR, where PQ = 6, QR = 15 and RP = 16

Let us see if each of these triangles is right by testing it with the Pythagorean Theorem.

*Triangle ABC*
AB^2 + BC^2 = 10^2 + 24^2 = 100 + 576 = 676
CA^2 = 26^2 = 676
=> AB^2 + BC^2 = CA^2
=> ABC is a right triangle.

*Triangle LMN*
LM^2 + MN^2 = 5^2 + 12^2 = 25 + 144 = 169
NL^2 = 13^2 = 169
=> LM^2 + MN^2 = NL^2
=> PQR is a right triangle.

*Triangle XYZ*
XY^2 + YZ^2 = 3^2 + 4^2 = 9 + 16 = 25
ZX^2 = 5^2 = 25
=> XY^2 + YZ^2 = ZX^2
=> XYZ is a right triangle.

*Triangle PQR*
PQ^2 + QR^2 = 6^2 + 15^2 = 36 + 225 = 261
RP^2 = 16^2 = 256
=> PQ^2 + QR^2 is not equal to RP^2

Therefore, *triangle PQR is not a right triangle*.

7. If there is a square whose diagonals are of length 2^(1/2) (i.e. the 'square root of two'), what is the length of each side of the square?

Answer: 1

In a square, all four sides are equal and both diagonals are equal as well. All four angles between sides are 90 degrees.

Let us now take into consideration a right triangle that has been formed with a diagonal and two sides of the square.
The diagonal = the hypotenuse = 2^(1/2)
Let the other two sides be 'x'.

Now, by the Pythagorean Theorem,

x^2 + x^2 = [2^(1/2)]^2
=> (2) (x^2) = 2
=> x^2 = 1
=> x = 1

And so, the length of each side of the square is *1*.

8. There is a triangle XYZ where XY is perpendicular to YZ and angle X = 70 degrees. What is the measure of angle Z?

Answer: 20 degrees

Since, XY is perpendicular to YZ,
=> Y = 90 degrees.

Now, by the angle sum property of a triangle,
angle X + angle Y + angle Z = 180 degrees
=> 70 degrees + 90 degrees + angle Z = 180 degrees
=> 160 degrees + angle Z = 180 degrees
=> *angle Z = 20 degrees*

And so that's your answer!

9. In a right triangle, the sum of the lengths of the two sides (not the hypotenuse) is *ALWAYS* greater than the length of the hypotenuse.

Answer: True

For *EVERY* triangle, this statement holds true: The sum of any two sides of a triangle is greater than the third side. This is known as the 'triangle inequality property'.

It isn't possible for a triangle where the sum of any two sides is not greater than the third side to exist. If you don't believe me, go ahead and try! ;-)

10. A right triangle, ABC, is given, where angle B = 90 degrees. The side AB is extended past angle B to form an exterior angle, X. What is the measure in degrees of angle X?

Answer: 90 degrees

By angle sum property of a triangle,

angle A + angle B + angle C = 180 degrees
=> angle A + angle B + 90 degrees = 180 degrees
=> angle A + angle B = 90 degrees

Now, exterior angle = sum of interior opposite angles.
=> angle X = angle A + angle C
=> *angle X = 90 degrees*

Source: Author achernar

This quiz was reviewed by FunTrivia editor CellarDoor before going online.
Any errors found in FunTrivia content are routinely corrected through our feedback system.

Right Triangles Trivia Quiz

A right triangle is nothing but a triangle which has one angle having a measure of 90 degrees. Test your knowledge about these, in a mixture of practical and theoretical questions. (Don't worry, no trigonometry involved!)

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>