Pregunta
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. \( \triangle P Q R \) is an isosceles triangle. \( \triangle P Q R \) is a right isosceles triangle. \( \triangle P Q R \) is an equilateral triangle. \( \triangle P Q R \) is a right triangle.
Ask by Tran Sandoval. in the United States
Jan 23,2025
Solución de tutoría real
Respuesta verificada por el tutor
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\( \triangle PQR \) is a right triangle.
Solución
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.**
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The Deep Dive
To classify triangle \( PQR \) using its vertices, first calculate the lengths of all three sides using the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). After plugging in the vertices, you'll find the side lengths. If two lengths are equal, then it’s isosceles; if one angle is 90 degrees, then it’s a right triangle. After calculating, you’ll determine the triangle type. Fun fact: Isosceles triangles have a special property where the angles opposite the equal sides are also equal, adding a twist to geometry’s magical world!
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