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

Answer:

To solve the problems related to square \( QRST \), we use the properties of a square:

• The diagonals of a square are equal in length and bisect each other at right angles.
• All sides of a square are equal.
• The diagonals of a square are congruent and bisect each other, so \( QV = VR = SV = VT \).
• In a square, if the side length is \( s \), the length of the diagonal \( d = s\sqrt{2} \).

Problem 28: Find \( SV \)

In square \( QRST \), the diagonals bisect each other. Given \( OV = 4 \) (assuming \( O \) is a typo and should be \( RV = 4 \) since \( RQ \) is a side? Wait, looking at the diagram, \( RQ \) is a side of the square, and \( RV \) is half of the diagonal? Wait, no, in a square, the diagonals bisect each other, so if \( RV = 4 \), then \( SV = RV = 4 \)? Wait, no, maybe \( RQ = 4 \) (side length of the square). Let's clarify:

If \( RQ \) is a side of the square with length \( 4 \), then the diagonal \( RT \) (and \( QS \)) can be found using the Pythagorean theorem. But first, let's assume the side length of the square is \( 4 \) (since \( RV \) is marked as \( 4 \), maybe \( RV \) is half the diagonal? Wait, no, in a square, the diagonals are equal and bisect each other, so \( RV = SV \) because the diagonals bisect each other. Wait, maybe the side length is \( 4 \). Let's re - examine:

In square \( QRST \), \( RQ \) is a side. If \( RQ = 4 \) (side length \( s = 4 \)), then the diagonal \( RT \) (and \( QS \)) is \( s\sqrt{2}=4\sqrt{2}\approx5.7 \), but the diagonals bisect each other, so \( RV = VT=\frac{RT}{2}\) and \( SV = VQ=\frac{QS}{2}\). Since \( RT = QS \) (diagonals of a square are equal), then \( RV = SV \). Wait, maybe the given length \( RV = 4 \), so \( SV = RV = 4 \)? But that seems too simple. Wait, maybe the side length is \( 4 \), so the diagonal is \( 4\sqrt{2}\approx5.7 \), and then \( SV \) is half of the diagonal? Wait, no, the diagonals intersect at \( V \), so \( V \) is the midpoint. So if the side length is \( 4 \), then the diagonal \( QS \) (or \( RT \)) is \( 4\sqrt{2}\), so \( SV=\frac{QS}{2}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\approx2.8 \)? Wait, I think I made a mistake. Let's start over.

Assume the side length of the square \( QRST \) is \( 4 \) (since \( RQ \) is labeled \( 4 \)). In a square, the diagonals are equal and bisect each other. The length of the diagonal \( d \) of a square with side length \( s \) is given by \( d = s\sqrt{2}\). So if \( s = 4 \), then \( d=4\sqrt{2}\approx5.7 \). The diagonals intersect at \( V \), so \( V \) divides each diagonal into two equal parts. Therefore, \( SV=\frac{d}{2}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\approx2.8 \)? Wait, no, maybe \( RQ \) is not the side. Wait, the diagram shows \( R \), \( Q \), \( T \), \( S \) as vertices of the square, with diagonals \( RT \) and \( QS \) intersecting at \( V \). If \( RV = 4 \), then since the diagonals of a square bisect each other, \( SV = RV = 4 \). But that would mean the diagonal \( RT = 8 \), so the side length \( s=\frac{8}{\sqrt{2}} = 4\sqrt{2}\approx5.7 \). But the problem says "Round to the nearest tenth". Wait, maybe the side length is \( 4 \), so the diagonal is \( 4\sqrt{2}\approx5.7 \), and \( SV \) is half of the diagonal, so \( SV=\frac{4\sqrt{2}}{2}=2\sqrt{2}\approx2.8 \)? No, I think I messed up. Let's check the properties again:

In a square, diagonals are congruent and bisect each other. So \( RV = SV \) (because diagonals bisect each other). If \( RV = 4 \) (from the diagram, \( RV \) is marked as \( 4 \)), then \( SV = 4 \). But that seems too easy. Wait, maybe the side length is \( 4 \), so the diagonal is \( 4\sqrt{2}\approx5.7 \), and \( SV \) is half of the diagonal, so \( SV=\frac{4\sqrt{2}}{2}=2\sqrt{2}\approx2.8 \). Wait, I think the confusion is between the side length and the half - diagonal. Let's assume the side length \( s = 4 \). Then the diagonal \( d=s\sqrt{2}=4\sqrt{2}\approx5.7 \). The diagonals intersect at \( V \), so \( V \) is the mid - point, so \( SV=\frac{d}{2}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\approx2.8 \). But maybe the given \( RV = 4 \) is the half - diagonal, so \( SV = RV = 4 \). This is confusing. Wait, maybe the problem has a typo, and \( RQ = 4 \) (side length). Then:

1. Side length of square \( s = 4 \).
2. Diagonal \( d=s\sqrt{2}=4\sqrt{2}\approx5.7 \).
3. Diagonals bisect each other, so \( SV=\frac{d}{2}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\approx2.8 \).

Problem 29: Find \( RT \)

If the side length \( s = 4 \), then using the Pythagorean theorem in triangle \( RQT \) (right - triangle with legs \( RQ = 4 \) and \( QT = 4 \)):

1. By Pythagorean theorem, \( RT^{2}=RQ^{2}+QT^{2}\).
2. Substitute \( RQ = 4 \) and \( QT = 4 \): \( RT^{2}=4^{2}+4^{2}=16 + 16=32 \).
3. Then \( RT=\sqrt{32}=4\sqrt{2}\approx5.7 \) (rounded to the nearest tenth).

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

First, we need to find the lengths of \( RV \), \( SV \), and \( RS \).

• We know \( RV = 2\sqrt{2}\approx2.8 \) (from Problem 28, if \( SV = 2\sqrt{2}\approx2.8 \)) and \( RS = 4 \) (side length of the square). Wait, no, if \( RV = 4 \) (from the diagram), then \( SV = 4 \), and \( RS = 4 \) (side length). Wait, let's use the correct values:

If \( RV = 4 \), \( SV = 4 \), and \( RS = 4 \) (side length), then the perimeter of \( \triangle RVS \) is \( RV+SV + RS=4 + 4+4 = 12 \). But that can't be right. Wait, no, if the side length \( s = 4 \), \( RV=\frac{RT}{2}=2\sqrt{2}\approx2.8 \), \( SV = 2\sqrt{2}\approx2.8 \), and \( RS = 4 \). Then the perimeter is \( 2.8+2.8 + 4=9.6 \).

Wait, let's start over with the correct approach:

Step 1: Determine the side length of the square

From the diagram, we can assume that the length of \( RQ \) (side of the square) is \( 4 \) (since it's marked as \( 4 \)).

Step 2: Find the length of the diagonal

For a square with side length \( s = 4 \), the length of the diagonal \( d\) (e.g., \( RT \) or \( QS \)) is given by the Pythagorean theorem:
\[
d=\sqrt{s^{2}+s^{2}}=\sqrt{2s^{2}}=s\sqrt{2}
\]
Substituting \( s = 4 \):
\[
d = 4\sqrt{2}\approx5.7
\]

Step 3: Find \( RV \) and \( SV \)

Since the diagonals of a square bisect each other, \( RV=SV=\frac{d}{2}\)
\[
RV = SV=\frac{4\sqrt{2}}{2}=2\sqrt{2}\approx2.8
\]

Step 4: Find the perimeter of \( \triangle RVS \)

The sides of \( \triangle RVS \) are \( RV\approx2.8 \), \( SV\approx2.8 \), and \( RS = 4 \) (side of the square).
The perimeter \( P\) of a triangle is the sum of its side lengths:
\[
P=RV + SV+RS
\]
Substituting the values:
\[
P\approx2.8 + 2.8+4=9.6
\]

Final Answers

1. \( SV=\boxed{2.8} \) (if side length is \( 4 \)) or \( SV = \boxed{4} \) (if \( RV = 4 \) and diagonals bisect each other). Wait, let's correct:

Assuming the side length of the square is \( 4 \):

• 28. \( SV=\frac{4\sqrt{2}}{2}=2\sqrt{2}\approx2.8 \), so \( SV=\boxed{2.8} \)
• 29. \( RT = 4\sqrt{2}\approx5.7 \), so \( RT=\boxed{5.7} \)
• 30. Perimeter of \( \triangle RVS \): \( 2.8 + 2.8+4 = 9.6 \), so perimeter \(=\boxed{9.6} \)