ISEE Upper Level Math : Geometry
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Example Questions
Example Question #5 : How To Find The Length Of The Side Of A Right Triangle
Note: Figure NOT drawn to scale.
Refer to the above diagram.
Find the length of 
First, find 
Since 
Example Question #8 : Triangles
Note: Figure NOT drawn to scale.
Refer to the above diagram. Evaluate 
By the Pythagorean Theorem,
Example Question #6 : How To Find The Length Of The Side Of A Right Triangle
A right triangle 
Insufficient information is given to answer the question.
The arcs intercepted by a right angle are both semicircles, so hypotenuse 
By the Pythagorean Theorem,
Example Question #1 : How To Find If Right Triangles Are Similar
Which is the greater quantity?
(a)
(b)
It is impossible to tell from the information given.
(a) is greater.
(b) is greater.
(a) and (b) are equal.
(a) and (b) are equal.
The hypotenuse 
Example Question #2 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7
If a right triangle has a base of 
To solve this problem, we must utilize the Pythagorean Theorom, which states that:
We know that the base is 
Next we evaluate the exponents:
Now we add them together:
Then, 
Example Question #3 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7
If a right triangle has a base of 
To solve this problem, we are going to use the Pythagorean Theorom, which states that 
We know that this particular right triangle has a base of 
Next, we evaluate the expoenents:
Then, 
To solve for 
Example Question #2 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
What is the hypotenuse of a right triangle with sides 5 and 8?
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According to the Pythagorean Theorem, the equation for the hypotenuse of a right triangle is 
Example Question #3 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7
In a right triangle, two sides have length 
By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let 
Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
In a right triangle, two sides have lengths 5 centimeters and 12 centimeters. Give the length of the hypotenuse.
This triangle has two angles of 45 and 90 degrees, so the third angle must measure 45 degrees; this is therefore an isosceles right triangle.
By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let 
Example Question #2 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
In a rectangle, the width is 6 feet long and the length is 8 feet long. If a diagonal is drawn through the rectangle, from one corner to the other, how many feet long is that diagonal?
Given that a rectangle has all right angles, drawing a diagonal will create a right triangle the legs are each 6 feet and 8 feet.
We know that in a 3-4-5 right triangle, when the legs are 3 feet and 4 feet, the hypotenuse will be 5 feet.
Given that the legs of this triangle are twice as long as those in the 3-4-5 triangle, it follows that the hypotense will also be twice as long.
Thus, the diagonal in through the rectangle creates a 6-8-10 triangle. 10 is therefore the length of the diagonal.
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