ISEE Middle Level

Triangles

Q: Do I need to memorize the Pythagorean Theorem for the ISEE?

Yes! The formula `a^2 + b^2 = c^2` is super important, especially if you are taking the Middle or Upper Level ISEE. It is the best tool for finding missing sides of right triangles.

Triangle area, Pythagorean theorem, angle sums, congruence, and similarity — excludes general area/perimeter of other shapes (see area-perimeter-composite)

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Imagine a perfect slice of pizza, a bright yellow yield sign, or the sail of a pirate ship. What do they all have in common? They are all triangles! 🍕⛵ Triangles are everywhere, and they are one of the most important shapes you will see on the ISEE.

A triangle is simply a shape with three straight sides and three inside angles. The absolute most important 'magic number' for triangles is 180. If you add up all the inside angles of any triangle in the universe—no matter how big, small, or weirdly shaped—they will always add up to exactly $18 0^{\circ}$ .

There are a few special types of triangles you should know. A 'right triangle' has one corner that looks like a perfect square $9 0^{\circ}$ angle (like the corner of your room). An 'equilateral triangle' has three equal sides and three equal angles, making it look like a perfect nacho chip.

On the ISEE, you will see triangles in both the Quantitative Reasoning and Mathematics Achievement sections. The test-makers love to ask you to find a missing angle, calculate the area, or compare the sizes of two different triangles. Don't worry! Once you learn a couple of secret rules, like the magic 180 rule and the area formula, triangle questions become super fun puzzles to solve! 🔺✨

Key Formula

The Area of a Triangle:

Area = \frac{1}{2} \cdot b \cdot h

(where

b

is the base and

h

is the straight-up-and-down height). Also, remember the magic angle rule:

Angle 1 + Angle 2 + Angle 3 = 18 0^{\circ}

Practice Questions

4 practice questions for ISEE Middle Level

Q1 Medium

A right triangle has legs of length 5 cm and 12 cm. What is the length of the hypotenuse?

A 11 cm

B 13 cm

C 15 cm

D 17 cm

Show Solution

Use the Pythagorean theorem: $a^{2} + b^{2} = c^{2}$ . Substitute the leg lengths: $5^{2} + 1 2^{2} = c^{2}$ , so $25 + 144 = 169$ . Therefore $c = 169 = 13$ cm.

Answer: B

Q2 Medium

Two triangles are similar. The smaller triangle has sides of 6 cm, 8 cm, and 10 cm. The longest side of the larger triangle is 25 cm. What is the length of the shortest side of the larger triangle?

A 12 cm

B 15 cm

C 18 cm

D 20 cm

Show Solution

The scale factor from the smaller triangle to the larger triangle is $\frac{25}{10} = \frac{5}{2}$ . Apply this scale factor to the shortest side: $6 \times \frac{5}{2} = 15$ cm.

Answer: B

Q3 Medium

Triangle

P QR

is a right triangle with the right angle at

Q

. If

P Q = 9

cm and

P R = 15

cm, what is the area of triangle

P QR

?

(Area of a triangle =

\frac{1}{2} bh

)

A 36 cm

^{2}

B 54 cm

^{2}

C 60 cm

^{2}

D 67.5 cm

^{2}

Show Solution

First, find the missing leg $QR$ using the Pythagorean theorem: $P Q^{2} + Q R^{2} = P R^{2}$ , so $81 + Q R^{2} = 225$ , which gives $Q R^{2} = 144$ and $QR = 12$ cm. The two legs of the right triangle serve as the base and height. Area $= \frac{1}{2} \times 9 \times 12 = 54$ cm $^{2}$ .

Answer: B

Q4 Medium

Triangle

A B C

has angles in the ratio 2 : 3 : 7. What is the measure of the largest angle?

A 60°

B 90°

C 105°

D 126°

Show Solution

The three angles of a triangle sum to 180°. The ratio 2 : 3 : 7 means the total number of parts is $2 + 3 + 7 = 12$ . Each part equals $180° \div 12 = 15°$ . The largest angle corresponds to the ratio of 7: $7 \times 15° = 105°$ .

Answer: C

Tips & Strategies

Look out for 'Pythagorean Triples'! The ISEE loves using right triangles with sides of $3 - 4 - 5$ and $5 - 12 - 13$ . If you memorize these sets, you can skip the math and save tons of time!
The height of a triangle MUST go straight up and down, making a $9 0^{\circ}$ angle with the base. Never use a slanted side as your height when finding the area!
If you are stuck on a Quantitative Comparison question about triangles, draw a quick sketch! Seeing the shapes on your scratch paper can help your brain figure out which one is bigger.

Common Mistakes

Watch out for forgetting the $\frac{1}{2}$ in the area formula! A lot of students just multiply base times height, but that gives you the area of a rectangle. Always remember to chop it in half!
Don't forget the magic number! Some students accidentally think triangle angles add up to $36 0^{\circ}$ . Remember, circles and squares use 360, but triangles are always $18 0^{\circ}$ .

Frequently Asked Questions

What exactly is a 'hypotenuse'?

The hypotenuse is just a fancy math word for the longest side of a right triangle! You can always find it easily because it is directly across from the $9 0^{\circ}$ square angle.

Do I need to memorize the Pythagorean Theorem for the ISEE?

Yes! The formula $a^{2} + b^{2} = c^{2}$ is super important, especially if you are taking the Middle or Upper Level ISEE. It is the best tool for finding missing sides of right triangles.

What if a triangle question is too hard and I don't know the answer?

Guess! The ISEE does not have a guessing penalty. If a triangle puzzle is taking too much time, pick your favorite letter, bubble it in, and move on. Never leave a question blank!

How do I know which side of the triangle is the 'base'?

The secret is that the base can be ANY side of the triangle! Just make sure that the 'height' you use is a straight line that drops from the opposite corner perfectly down to that base.

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