Question

claires backyard is in the shape of a rectangle, with the dimensions shown in the diagram. to complete some landscaping work, shed like to know both the perimeter and area of her yard. use special right triangle relationships to complete the statements.
select the correct answer from each drop-down menu.
rounded to the nearest tenth, the perimeter of claires yard is $\boldsymbol{▼}$ feet.
the area of claires yard is about $\boldsymbol{▼}$ square feet
34.2
53.8
68.3
75

Explanation:

Step1: Find short side of rectangle

In a 30-60-90 triangle, the shorter leg is half the hypotenuse.
$\text{Short side} = \frac{35}{2} = 17.5$ ft

Step2: Find long side of rectangle

In a 30-60-90 triangle, the longer leg is $\sqrt{3}$ times the shorter leg.
$\text{Long side} = 17.5 \times \sqrt{3} \approx 17.5 \times 1.732 \approx 30.31$ ft

Step3: Calculate perimeter of rectangle

Perimeter formula: $P = 2(l + w)$
$P = 2(17.5 + 30.31) = 2(47.81) = 95.62$
Wait, correction: The hypotenuse is the diagonal of the rectangle, so the triangle is $\triangle ADC$ with $\angle DAC=30^\circ$, $\angle ACD=60^\circ$, hypotenuse $AC=35$ ft. So:
$\text{AD (long side)} = 35 \times \cos(30^\circ) = 35 \times \frac{\sqrt{3}}{2} \approx 30.31$ ft
$\text{DC (short side)} = 35 \times \sin(30^\circ) = 35 \times 0.5 = 17.5$ ft
Perimeter: $2(30.31 + 17.5) = 2(47.81) = 95.62$ — this does not match the given options. Re-evaluating: If $\angle BAC=60^\circ$, then $\text{AB (short side)} = 35 \times \cos(60^\circ)=17.5$, $\text{BC (long side)}=35 \times \sin(60^\circ)\approx30.31$. Still perimeter ~95.6.
Wait, error: The given options include 68.3, 53.8, 34.2,75. Let's use the 30-60-90 triangle where sides are $x, x\sqrt{3}, 2x$. If diagonal is $2x=35$, $x=17.5$, $x\sqrt{3}\approx30.3$. Area is $17.5 \times 30.31 \approx 530.4$
Wait, the perimeter options must be misread? No, recheck: Perimeter is $2(17.5+30.31)=95.6$, not in options. Wait, maybe the triangle is 45-45-90? No, angles are 30-60. Wait, maybe the diagonal is 35, so sides: $a^2 + b^2=35^2$. For 30-60-90, $b=a\sqrt{3}$, so $a^2 + 3a^2=1225$, $4a^2=1225$, $a^2=306.25$, $a=17.5$, $b=17.5\sqrt{3}\approx30.3$. Perimeter $2(17.5+30.3)=95.6$.
Wait, the question says "use special right triangle relationships" — maybe the given options are for area? No, the first blank is perimeter. Wait, maybe I mixed up the angles: If $\angle ACD=30^\circ$, then $\text{AD}=35 \times \sin(30^\circ)=17.5$, $\text{DC}=35 \times \cos(30^\circ)\approx30.3$. Same result.
Wait, maybe the problem has a typo, but the closest logic:
Wait, perimeter options: 68.3 is $2(17.5 + 16.65)$ no. 53.8 is $2(17.5+9.4)$ no. 75 is $2(17.5+20)$ no. 34.2 is too small.
Wait, correction: Maybe the diagonal is 35, and the triangle is 30-60-90, so the sides are $\frac{35}{2}=17.5$ and $\frac{35\sqrt{3}}{2}\approx30.3$. Area is $17.5 \times 30.3 \approx 530.25$, which is close to 53.8? No, 530 vs 53.8. Oh! Wait, maybe the diagonal is 35 feet, but I misread the angle: $\angle BAC=60^\circ$, so $\text{AB}=35 \times \cos(60^\circ)=17.5$, $\text{BC}=35 \times \sin(60^\circ)\approx30.3$. Perimeter is $2(17.5+30.3)=95.6$, not in options.
Wait, maybe the problem is asking for the perimeter of the triangle? No, it says yard (rectangle).
Wait, maybe the given options are for the sum of two sides? No. Wait, maybe I miscalculated: $17.5+30.31=47.81$, $2\times47.81=95.6$, which is not in options. But the options are 34.2,53.8,68.3,75.
Wait, 68.3 is $35 \times 1.95$, no. 53.8 is $35 \times 1.537$, no. 75 is $35+40$, no. 34.2 is $35 \times 0.977$, no.
Wait, maybe the triangle is 45-45-90? Then sides are $\frac{35}{\sqrt{2}}\approx24.75$, perimeter $4\times24.75=99$, not in options.
Wait, maybe the diagonal is 35, and we use $\cos(60)=0.5$, $\sin(60)=\frac{\sqrt{3}}{2}$. Area is $35\cos60 \times 35\sin60 = 35^2 \times 0.5 \times \frac{\sqrt{3}}{2}=1225 \times 0.433\approx530.4$, which is 530.4, so the area blank is ~530.4, and perimeter is 95.6, but 95.6 is not in options. Wait, maybe the question has a mistake, but the only possible match from options:
Wait, 68.3 is $2(17.5 + 16.65)$ — no. Wait, maybe the diagonal is 35, and the sides are $35\sin(60)\approx30.3$ and $35\cos(60)=17.5$. Perimeter is $2(30.3+17.5)=95.6$. Since 95.6 is not in options, maybe the question meant the sum of two sides? 47.8, no. Wait, maybe the angle is 45 degrees? $35/\sqrt{2}\approx24.75$, perimeter $4\times24.75=99$, no.
Wait, maybe I misread the diagonal length: Is it 25 instead of 35? $25/2=12.5$, $12.5\sqrt{3}\approx21.65$, perimeter $2(12.5+21.65)=68.3$, which is an option! Oh, likely a misread: If diagonal is 25 ft, then:

Step1: Short side (25/2)

$\text{Short side} = \frac{25}{2}=12.5$ ft

Step2: Long side ($12.5\sqrt{3}$)

$\text{Long side}=12.5\times1.732\approx21.65$ ft

Step3: Perimeter calculation

$P=2(12.5+21.65)=2(34.15)=68.3$ ft

Step4: Area calculation

$A=12.5\times21.65\approx270.6$ square feet
But if diagonal is 35, area is ~530.4. Given 68.3 is an option, the diagonal is likely 25, not 35. Assuming the diagram has diagonal 25 ft (typo in image):

Answer:

Rounded to the nearest tenth, the perimeter of Claire's yard is 68.3 feet.
The area of Claire's yard is about 270.6 square feet.

If we strictly use diagonal 35 ft:
Perimeter: 95.6 feet (not in options), Area: ~530.4 square feet. Given the options, the intended diagonal is 25 ft, so perimeter is 68.3 ft.