olamiami.performancematterscom/ola/olajsp?clientCode=fiMiamiDade* Performance Matters Welcome, ZIONI Question 28 of 50 Submit Test M-DCPS - MYA-Math-Geometry - CBT - 2024-2025 -... Question 1-28 A triangle $$ P Q R $$ has vertices $$ P(-1,3), Q(9,-1) $$ and $$ R(-3,-2) $$. Identify the type of triangle and select the correct statement. $$ riangle P Q R $$ is an isosceles triangle. $$ riangle P Q R $$ is a right isosceles triangle. $$ riangle P Q R $$ is an equilateral triangle. $$ riangle P Q R $$ is a right triangle.

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olamiami.performancematterscom/ola/olajsp?clientCode=fiMiamiDade* Performance Matters Welcome, ZIONI Question 28 of 50 Submit Test M-DCPS - MYA-Math-Geometry - CBT - 2024-2025 -... Question 1-28 A triangle $$ P Q R $$ has vertices $$ P(-1,3), Q(9,-1) $$ and $$ R(-3,-2) $$. Identify the type of triangle and select the correct statement. $$ 	riangle P Q R $$ is an isosceles triangle. $$ 	riangle P Q R $$ is a right isosceles triangle. $$ 	riangle P Q R $$ is an equilateral triangle. $$ 	riangle P Q R $$ is a right triangle.

UpStudy AI · Accepted Answer

To determine the type of triangle $$ \triangle PQR $$ with vertices $$ P(-1,3) $$, $$ Q(9,-1) $$, and $$ R(-3,-2) $$, we'll calculate the lengths of its sides and analyze their relationships.

### Step 1: Calculate the Lengths of the Sides

1. **Length of $$ PQ $$:**
   $$
   PQ = \sqrt{(9 - (-1))^2 + (-1 - 3)^2} = \sqrt{10^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116} \approx 10.77
   $$

2. **Length of $$ QR $$:**
   $$
   QR = \sqrt{(-3 - 9)^2 + (-2 - (-1))^2} = \sqrt{(-12)^2 + (-1)^2} = \sqrt{144 + 1} = \sqrt{145} \approx 12.04
   $$

3. **Length of $$ RP $$:**
   $$
   RP = \sqrt{(-1 - (-3))^2 + (3 - (-2))^2} = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.39
   $$

### Step 2: Analyze the Side Lengths

- **Scalene vs. Isosceles vs. Equilateral:**
  - All three sides have different lengths ($$ PQ \neq QR \neq RP $$), so the triangle is **scalene**. 
  - However, none of the provided options mention a scalene triangle.

- **Checking for a Right Triangle:**
  - To determine if the triangle is a right triangle, we can use the Pythagorean theorem. Check if the sum of the squares of the two shorter sides equals the square of the longest side.
  
  $$
  PQ^2 + RP^2 = 116 + 29 = 145 \approx QR^2
  $$
  
  - Since $$ PQ^2 + RP^2 \approx QR^2 $$, $$ \triangle PQR $$ satisfies the Pythagorean theorem, indicating that it is a **right triangle**.

### Conclusion

Among the provided options, the correct statement is:

**$$ \triangle P Q R $$ is a right triangle.**
$$ \triangle PQR $$ is a right triangle.

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