Question

two similar triangles have perimeters of 24 cm and 36 cm respectively. if a side of the smaller triangle is 8 cm, what is the length of the corresponding side in the larger triangle?
○ a. 16 cm
○ b. 18 cm
○ c. 10 cm
○ d. 12 cm

in a triangle, one leg is 12 cm, and the other leg is 16 cm. what is the length of the hypotenuse?
○ a. 20 cm
○ b. 22 cm
○ c. 24 cm
○ d. 18 cm

which of the following is not a criterion for triangle similarity?
○ a. ssa
○ b. sas
○ c. sss
○ d. aa

Explanation:

First Question:

Step1: Find the ratio of perimeters

The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides. The perimeter of the smaller triangle is 24 cm and the larger is 36 cm. So the ratio of perimeters is $\frac{24}{36}=\frac{2}{3}$. Wait, no, actually, the ratio of the smaller to larger is $\frac{24}{36}=\frac{2}{3}$, but we need the ratio of smaller side to larger side, which should be equal to the ratio of their perimeters. Let the length of the corresponding side in the larger triangle be $x$. So $\frac{8}{x}=\frac{24}{36}$.

Step2: Solve for $x$

Simplify $\frac{24}{36}=\frac{2}{3}$. So we have $\frac{8}{x}=\frac{2}{3}$. Cross - multiply: $2x = 8\times3=24$. Then $x=\frac{24}{2}=12$? Wait, no, wait. Wait, the perimeter of smaller is 24, larger is 36. So the scale factor from smaller to larger is $\frac{36}{24}=\frac{3}{2}$. So the corresponding side in larger triangle is $8\times\frac{3}{2}=12$? Wait, no, let's do it again. Let the ratio of smaller to larger perimeter be $k=\frac{24}{36}=\frac{2}{3}$. Then the ratio of corresponding sides is also $k$. So if the smaller side is 8, then $8 = k\times x$, where $x$ is the larger side. So $x=\frac{8}{k}=\frac{8}{\frac{2}{3}}=8\times\frac{3}{2}=12$. Wait, but let's check the options. Option d is 12 cm. Wait, but maybe I made a mistake. Wait, the perimeter ratio is 24:36 = 2:3. So the side ratio is also 2:3. So smaller side : larger side = 2:3. So 8 : $x$ = 2:3. So $2x = 24$, $x = 12$. So the answer is d. 12 cm.

Step1: Recall Pythagorean theorem

For a right - triangle, the Pythagorean theorem states that $c^{2}=a^{2}+b^{2}$, where $c$ is the hypotenuse and $a$ and $b$ are the legs. Here, $a = 12$ cm and $b = 16$ cm.

Step2: Calculate $a^{2}+b^{2}$

$a^{2}=12^{2}=144$, $b^{2}=16^{2}=256$. Then $a^{2}+b^{2}=144 + 256=400$.

Step3: Find $c$

Since $c^{2}=400$, then $c=\sqrt{400}=20$ cm.

• The criteria for triangle similarity are:
• AA (Angle - Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
• SAS (Side - Angle - Side) similarity: If the ratio of two sides of one triangle is equal to the ratio of two sides of another triangle and the included angles are equal, the triangles are similar.
• SSS (Side - Side - Side) similarity: If the ratio of all three corresponding sides of two triangles is equal, the triangles are similar.
• SSA (Side - Side - Angle) is not a criterion for triangle similarity. In fact, SSA does not guarantee congruence (and thus not similarity) because there can be two different triangles with the same SSA measurements (the ambiguous case in triangle congruence).

Answer:

d. 12 cm

Second Question: