Chapter 4: Q17. (page 151)

A, B, C, and D are noncoplanar. $Δ ABC, Δ ACD$ and $Δ ABD$ are equilateral. X and Y are midpoints of $\bar{AC} and \bar{AD}$ . Z is a point on $\bar{AB}$ . What kind of triangle is $Δ XYZ$ ? Explain.

Short Answer

Expert verified

$Δ X Y Z$ is anisosceles triangle.

Step by step solution

Step 1. Description of step.

Consider $Δ X A Z$ and $Δ Y A Z$ . From the given information, $\bar{A Z} ≅ \bar{A Z}$ , $∠ X A Z ≅ ∠ Y A Z$ and $\bar{A Y} ≅ \bar{A X}$ then by SAS postulate, $Δ X A Z ≅ Δ Y A Z$ .

Step 2. Description of step.

By side angle side postulate $Δ A B C ≅ Δ A C D$ as they are equilateral triangles. Hence, the corresponding parts $\bar{X Y}$ and $\bar{Y Z}$ of congruent triangles $Δ X A Z$ and $Δ Y A Z$ are congruent.

Therefore, $Δ X Y Z$ is an isosceles triangle.

Step 3. Write a paragraph proof.

$Δ X A Z ≅ Δ Y A Z$ by SAS postulate. $Δ A B C ≅ Δ A C D$ by SAS postulate. The corresponding parts $\bar{X Y}$ and $\bar{Y Z}$ of congruent triangles $Δ X A Z$ and $Δ Y A Z$ are congruent.

Therefore, $Δ X Y Z$ is an isosceles triangle.

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A, B, C, and D are noncoplanar. $Δ ABC, Δ ACD$ and $Δ ABD$ are equilateral. X and Y are midpoints of $\bar{AC} and \bar{AD}$ . Z is a point on $\bar{AB}$ . What kind of triangle is $Δ XYZ$ ? Explain.

Short Answer

Step by step solution

Step 1. Description of step.

Step 2. Description of step.

Step 3. Write a paragraph proof.

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A, B, C, and D are noncoplanar. ΔABC,ΔACDand ΔABDare equilateral. X and Y are midpoints of AC¯andAD¯. Z is a point on AB¯. What kind of triangle is ΔXYZ? Explain.

Short Answer

Step by step solution

Step 1. Description of step.

Step 2. Description of step.

Step 3. Write a paragraph proof.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Calculus

Theoretical and Mathematical Physics

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Mechanics Maths

Study anywhere. Anytime. Across all devices.

A, B, C, and D are noncoplanar. $Δ ABC, Δ ACD$ and $Δ ABD$ are equilateral. X and Y are midpoints of $\bar{AC} and \bar{AD}$ . Z is a point on $\bar{AB}$ . What kind of triangle is $Δ XYZ$ ? Explain.