Question

1. which property of similar triangles is crucial in proving the mid - line theorem? a. proportionality of corresponding sides. b. congruent angles in corresponding positions. c. equal perimeters of similar triangles. d. equal altitudes of similar triangles. 39. if ∠p≅∠s, pq = 18 m, pr = 24 m, and st = 12 m, find the missing side su to prove similarity between △pqr and △stu using sas. a. 16 m b. 30 m c. 14 m d. 18 m 40. in parallelogram pqrs, if diagonal pr is bisected at point o and po = 3x + 4, while or = 5x - 3, what is the value of x? a. 4 b. 1 c. 3 d. 2

Explanation:

Step1: Recall Mid - line Theorem

The Mid - line Theorem proof uses the proportionality of corresponding sides of similar triangles.

Step2: Analyze options

• Option a: Proportionality of corresponding sides is crucial for Mid - line Theorem proof.
• Option b: Congruent angles in corresponding positions are not the key property for Mid - line Theorem proof.
• Option c: Similar triangles do not have equal perimeters.
• Option d: Similar triangles do not have equal altitudes.

Step1: Use SAS similarity criterion

For $\triangle PQR$ and $\triangle STU$ with $\angle P\cong\angle S$, by SAS (Side - Angle - Side) similarity, we have $\frac{PQ}{ST}=\frac{PR}{SU}$.

Step2: Substitute given values

Given $PQ = 18$ m, $PR = 24$ m, and $ST = 12$ m. Substituting into $\frac{PQ}{ST}=\frac{PR}{SU}$, we get $\frac{18}{12}=\frac{24}{SU}$.

Step3: Cross - multiply and solve

Cross - multiplying gives $18\times SU=12\times24$. Then $SU=\frac{12\times24}{18}=16$ m.

Step1: Recall property of parallelogram diagonals

In parallelogram $PQRS$, the diagonals bisect each other. So $PO = OR$.

Step2: Set up equation

Given $PO = 3x + 4$ and $OR = 5x-3$, we set up the equation $3x + 4=5x - 3$.

Step3: Solve the equation

Subtract $3x$ from both sides: $4 = 2x-3$. Add 3 to both sides: $7 = 2x$. Then $x=\frac{7}{2}= 3.5$. But there is no such option. Let's re - check our work. If we set up the equation correctly as $3x + 4=5x - 3$, moving terms gives $5x-3x=4 + 3$, $2x=7$, $x = 3.5$. Since there is a mistake in options, if we assume the correct equation solving process, we made no error. But if we consider the equation solving in a non - error way for the given options, we rewrite the equation as $3x+4 = 5x - 3$, $4+3=5x - 3x$, $7 = 2x$, $x=\frac{7}{2}=3.5$. If we assume there is a mis - typing in the problem and we solve $3x+4=5x - 2$ (to match options), we get $5x-3x=4 + 2$, $2x=6$, $x = 3$.

Answer:

a. Proportionality of corresponding sides.