Question

given: $overline{ys}congoverline{yu}$, $overline{ys}perpoverline{rt}$, $overline{yu}perpoverline{rv}$, y is the mid - point of $overline{tv}$. prove: $\triangle trv$ is isosceles. 4. $overline{ys}congoverline{yu}$, y is the mid - point of $overline{tv}$. 5. $overline{vy}congoverline{ty}$ 6. $\triangle vuycong\triangle tsy$ 7. click here to insert

Explanation:

Step1: Identify given facts

We are given side - congruence, perpendicularity, and mid - point information.

Step2: Use mid - point property

$Y$ is mid - point of $\overline{TV}$, so $\overline{VY}\cong\overline{TY}$.

Step3: Note right - angle formation

Perpendicular lines $\overline{YS}\perp\overline{RT}$ and $\overline{YU}\perp\overline{RV}$ give right angles.

Step4: Prove triangle congruence

Use HL criterion with given side and right - angle information to show $\triangle VUY\cong\triangle TSY$.

Step5: Get congruent sides of large triangle

By CPCTC, $\overline{RV}\cong\overline{RT}$.

Step6: Apply isosceles triangle definition

Since $\overline{RV}\cong\overline{RT}$, $\triangle TRV$ is isosceles.

Answer:

1. Statements:

• Given: $\overline{YS}\cong\overline{YU}$, $\overline{YS}\perp\overline{RT}$, $\overline{YU}\perp\overline{RV}$, $Y$ is the mid - point of $\overline{TV}$
• $\overline{VY}\cong\overline{TY}$ (Definition of mid - point)
• $\angle YSV = 90^{\circ}$, $\angle YUV=90^{\circ}$ (Definition of perpendicular lines)
• $\triangle VUY$ and $\triangle TSY$:
• $\overline{YS}\cong\overline{YU}$ (Given)
• $\angle YSV=\angle YUV$ (Both are right angles)
• $\overline{VY}\cong\overline{TY}$ (Definition of mid - point)
• So, $\triangle VUY\cong\triangle TSY$ (HL congruence criterion)
• Then, $\overline{RV}\cong\overline{RT}$ (Corresponding parts of congruent triangles are congruent)
• Since $\overline{RV}\cong\overline{RT}$, $\triangle TRV$ is isosceles (Definition of isosceles triangle: a triangle with two congruent sides)

1. Reasons for each step:

• Step 1: Given information
• Step 2: By the definition of a mid - point, if $Y$ is the mid - point of $\overline{TV}$, then the two segments formed are congruent.
• Step 3: Perpendicular lines form right angles.
• Step 4: We list the congruent parts for the two right - angled triangles. The Hypotenuse - Leg (HL) congruence criterion states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, the two right - triangles are congruent. Here, $\overline{VY}$ and $\overline{TY}$ are the hypotenuses and $\overline{YS}$ and $\overline{YU}$ are the legs.
• Step 5: Corresponding parts of congruent triangles are congruent (CPCTC).
• Step 6: A triangle is isosceles if it has two congruent sides. Since $\overline{RV}\cong\overline{RT}$, $\triangle TRV$ is isosceles.