How to find if triangles are similar

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ISEE Upper Level Quantitative Reasoning › How to find if triangles are similar

Questions 1 - 2

$\Delta ABC$ and $\Delta DEF$ are right triangles, with right angles $\angle B , \angle E$ , respectively. $m \angle A = m\angle D = 45^{\circ }$ and .

Which is the greater quantity?

(a)

(b)

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Explanation

Each right triangle is a $45 ^{\circ }-45 ^{\circ }-90 ^{\circ }$ triangle, making each triangle isosceles by the Converse of the Isosceles Triangle Theorem.

Since $\angle B$ and $\angle E$ are the right triangles, the legs are $\overline{AB}, \overline{BC}, \overline{DE}, \overline{EF}$ , and the hypotenuses are $\overline{AC}, \overline{DF}$ .

By the $45 ^{\circ }-45 ^{\circ }-90 ^{\circ }$ Theorem, $AB \sqrt{2} = AC$ and $EF \sqrt{2} = DF$ .

, so $AB \sqrt{2} > EF \sqrt{2}$ and subsequently, .

Untitled

Figure NOT drawn to scale.

In the above figure, $\angle ABC$ is a right angle.

What is the ratio of the area of $\bigtriangleup AXB$ to that of $\bigtriangleup BXC$ ?

144 to 25

169 to 25

12 to 5

13 to 5

Explanation

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 $\bigtriangleup AXB$ to $\bigtriangleup BXC$ can be found by determining the ratio of one pair of corresponding sides; we will use the short leg of each, $\overline{XB}$ and $\overline{XC}$ .