Question

rectangle pqrs is shown with its diagonals, \\(\overline{pr}\\) and \\(\overline{qs}\\).

what type of triangle is \\(\delta str\\)?

• right triangle
• equilateral triangle
• isosceles triangle
• scalene triangle

Explanation:

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<reasoning>

Identify properties of rectangle diagonals

Using the Properties of Rectangles knowledge point
\[

$$\begin{aligned} &PR = QS \quad \text{(Diagonals are equal in length)}\\ &PT = TR = \frac{1}{2}PR \quad \text{(Diagonals bisect each other)}\\ &ST = TQ = \frac{1}{2}QS \quad \text{(Diagonals bisect each other)} \end{aligned}$$

\]

Determine relationship between segment lengths

Using the Properties of Rectangles knowledge point
\[

$$\begin{aligned} &TR = \frac{1}{2}PR\\ &ST = \frac{1}{2}QS\\ &PR = QS \implies ST = TR \end{aligned}$$

\]

Classify the triangle

Using the Properties of Rectangles knowledge point
\[

$$\begin{aligned} &ST = TR \implies \triangle STR \text{ has two equal sides}\\ &\text{Therefore, } \triangle STR \text{ is an isosceles triangle} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>right triangle</mcq-option>
<mcq-option>equilateral triangle</mcq-option>
<mcq-correct>isosceles triangle</mcq-correct>
<mcq-option>scalene triangle</mcq-option>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Identify properties of rectangle diagonals

Using the Properties of Rectangles knowledge point
\[

$$\begin{aligned} &PR = QS \quad \text{(Diagonals are equal in length)}\\ &PT = TR = \frac{1}{2}PR \quad \text{(Diagonals bisect each other)}\\ &ST = TQ = \frac{1}{2}QS \quad \text{(Diagonals bisect each other)} \end{aligned}$$

\]

Determine relationship between segment lengths

Using the Properties of Rectangles knowledge point
\[

$$\begin{aligned} &TR = \frac{1}{2}PR\\ &ST = \frac{1}{2}QS\\ &PR = QS \implies ST = TR \end{aligned}$$

\]

Classify the triangle

Using the Properties of Rectangles knowledge point
\[

$$\begin{aligned} &ST = TR \implies \triangle STR \text{ has two equal sides}\\ &\text{Therefore, } \triangle STR \text{ is an isosceles triangle} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>right triangle</mcq-option>
<mcq-option>equilateral triangle</mcq-option>
<mcq-correct>isosceles triangle</mcq-correct>
<mcq-option>scalene triangle</mcq-option>
</answer>

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