Question

for exercises 28–30, find each length and value for square qrst. round to the nearest tenth. see example 5 28. sv 29. rt 30. perimeter of △rvs

Explanation:

Problem 28: Find \( SV \)

In a square, the diagonals are equal in length and bisect each other. So, the diagonals of square \( QRST \) (i.e., \( RT \) and \( QS \)) bisect each other at point \( V \). First, we can find the length of the diagonal using the side length of the square. The side length of the square (e.g., \( RQ \)) is given as 4.

Step 1: Find the length of diagonal \( QS \) (or \( RT \))

In a square, the length of the diagonal \( d \) is related to the side length \( s \) by the formula \( d = s\sqrt{2} \). Here, \( s = 4 \), so:
\[
d = 4\sqrt{2}
\]

Step 2: Find \( SV \)

Since the diagonals bisect each other, \( SV=\frac{1}{2}QS \). Substituting \( QS = 4\sqrt{2} \):
\[
SV=\frac{1}{2}\times4\sqrt{2}=2\sqrt{2}\approx2\times1.414 = 2.828\approx2.8
\]

Problem 29: Find \( RT \)

As mentioned earlier, in a square with side length \( s \), the diagonal \( d = s\sqrt{2} \). The side length \( s = 4 \).

Step 1: Apply the diagonal formula

Using \( d = s\sqrt{2} \) with \( s = 4 \):
\[
RT = 4\sqrt{2}\approx4\times1.414 = 5.656\approx5.7
\]

Problem 30: Find the perimeter of \( \triangle RVS \)

First, we know the lengths of the sides of \( \triangle RVS \). From the square, \( RV = SV \) (since diagonals bisect each other) and \( RS \) is a side of the square (length 4). We already found \( RV = SV = 2\sqrt{2}\approx2.8 \) and \( RS = 4 \).

Step 1: Identify the lengths of the sides of \( \triangle RVS \)

• \( RV\approx2.8 \)
• \( SV\approx2.8 \)
• \( RS = 4 \)

Step 2: Calculate the perimeter

The perimeter \( P \) of a triangle is the sum of its side lengths:
\[
P=RV + SV+RS\approx2.8 + 2.8+4 = 9.6
\]

Answer:

s:

1. \( \boldsymbol{SV\approx2.8} \)
2. \( \boldsymbol{RT\approx5.7} \)
3. \( \boldsymbol{\text{Perimeter of } \triangle RVS\approx9.6} \)