How to find if triangles are similar - ISEE Upper Level (grades 9-12) Quantitative Reasoning
Card 0 of 2
and 
are right triangles, with right angles 
, respectively. 
and 
.
Which is the greater quantity?
(a) 
(b) 
Which is the greater quantity?
(a)
(b)
Each right triangle is a 
triangle, making each triangle isosceles by the Converse of the Isosceles Triangle Theorem.
Since 
and 
are the right triangles, the legs are 
, and the hypotenuses are 
.
By the Theorem, 
and 
.
, so 
and subsequently, 
.
Each right triangle is a
Since
By the
Compare your answer with the correct one above
Figure NOT drawn to scale.
In the above figure, 
is a right angle.
What is the ratio of the area of 
to that of 
?
Figure NOT drawn to scale.
In the above figure,
What is the ratio of the area of
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller, similar triangles.
The similarity ratio of 
to 
can be found by determining the ratio of one pair of corresponding sides; we will use the short leg of each, 
and 
.
is also the long leg of 
, so its length can be found using the Pythagorean Theorem:
The similarity ratio is therefore
.
The ratio of the areas is the square of this ratio:
- that is, 144 to 25.
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller, similar triangles.
The similarity ratio of
The similarity ratio is therefore
The ratio of the areas is the square of this ratio:
Compare your answer with the correct one above