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Advanced Geometry : Trapezoids

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Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Find the length of diagonal $\overline{AB}$ of the trapezoid.

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Possible Answers:

$AB = \sqrt{41+10\cdot \sqrt{7}} \approx 8.21$

AB = 8 + 6 = 14

$AB = \sqrt{34}$

AB = 4 + 5 = 9

$AB = \sqrt{73}$

Correct answer:

$AB = \sqrt{41+10\cdot \sqrt{7}} \approx 8.21$

Explanation:

1) The diagonal $\overline{AB}$ can be found from $\bigtriangleup BAE$ by using the Pythagorean Theorem.

2) The length of the base of $\bigtriangleup CAD$ , $\overline{AD}$ has to be found because AE = AD + DE is the length of the base of $\bigtriangleup BAE$ .

3) AE = 5 + AD .

4) Using the Pythagorean Theorem on $\bigtriangleup CAD$ to find $m(\overline{AD})$ ,

$4^{2} = 3^{2} + (AD)^{2}$

$4^{2} - 3^{2} = (AD)^{2}$

$16 - 9 = (AD)^{2}$

$7 = (AD)^{2}$

$\sqrt{7} = AD$

5) Using the Pythagorean Theorem on $\bigtriangleup BAE$ to find $m(\overline{AB})$ ,

$(AB)^{2} = (5 + \sqrt{7})^{2} + 3^{2}$

$= 5^{2} + 2(5\cdot \sqrt{7}) + (\sqrt{7})^{2} + 9$

$= 25 + 10\cdot \sqrt{7} + 7 + 9$

$= 41 + 10\sqrt{7}$

$\sqrt{(AB)^{2}} = \sqrt{41 + 10\sqrt{7}}$

$AB = \sqrt{41 + 10\sqrt{7}} \approx 8.21$

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Example Question #3 : How To Find The Length Of The Diagonal Of A Trapezoid

Trapezoid

Figure NOT drawn to scale.

Refer to the above diagram, which shows Trapezoid TRAP with diagonal $\overline{TA}$ . To the nearest whole number, give the length of $\overline{TA}$ .

Possible Answers:

104

Correct answer:

Explanation:

To illustrate how to determine the correct length, draw a perpendicular segment from to $\overline{PA}$ , calling the point of intersection .

Trapezoid

$\overline{RX}$ divides the trapezoid into Rectangle TRXP and right triangle $\bigtriangleup RXA$ .

Opposite sides of a rectangle are congruent, so PX = TR = 40 .

$m \angle TRA = 150 ^{\circ }$ . The two angles of a trapezoid along the same leg - in particular, $\angle TRA$ and $\angle RAP$ - are supplementary, so

$m \angle RAP = 180 ^{\circ } - m \angle TRA$

$= 180 ^{\circ } - 150 ^{\circ }$

$= 30 ^{\circ }$

By the 30-60-90 Triangle Theorem,

$XA = RX\cdot \sqrt{3}$

Opposite sides of a rectangle are congruent, so RX= TP = 30 , and

$XA = 30 \sqrt{3}$

$\approx 30 \cdot 1.7321$

$\approx 52$

$PA = PX + XA \approx 40+ 52 \approx 92$

$\overline{TA}$ is the hypotenuse of right triangle $\bigtriangleup TPA$ , so by the Pythagorean Theorem, its length can be calculated to be

$TA = \sqrt{ (TP )^{2}+ (PA)^{2}}$

Set TP = 30 and PA = 92 :

$TA = \sqrt{ 30 ^{2}+ 92^{2}}$

$= \sqrt{ 900 + 8,464 }$

$= \sqrt{9,364 }$

$\approx 97$

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Example Question #4 : How To Find The Length Of The Diagonal Of A Trapezoid

Trapezoid

Figure NOT drawn to scale.

Refer to the above diagram, which shows Trapezoid TRAP with diagonal $\overline{TA}$ . To the nearest whole number, give the length of $\overline{TA}$ .

Possible Answers:

104

Correct answer:

Explanation:

To illustrate how to determine the correct length, draw a perpendicular segment from to $\overline{PA}$ , calling the point of intersection .

Trapezoid

$\overline{RX}$ divides the trapezoid into Rectangle TRXP and right triangle $\bigtriangleup RXA$ .

Opposite sides of a rectangle are congruent, so PX = TR = 40 .

$m \angle TRA = 120 ^{\circ }$ . The two angles of a trapezoid along the same leg - in particular, $\angle TRA$ and $\angle RAP$ - are supplementary, so

$m \angle RAP = 180 ^{\circ } - m \angle TRA$

$= 180 ^{\circ } - 120 ^{\circ }$

$= 60 ^{\circ }$

By the 30-60-90 Triangle Theorem,

$XA = \frac{RX}{ \sqrt{3} }$

Opposite sides of a rectangle are congruent, so RX= TP = 30 , and

$XA = \frac{RX}{ \sqrt{3} }$

$\approx \frac{30 }{1.7321}$

$\approx 17.3$

$PA = PX + XA \approx 40+ 17.3 \approx 57.3$

$\overline{TA}$ is the hypotenuse of right triangle $\bigtriangleup TPA$ , so by the Pythagorean Theorem, its length can be calculated to be

$TA = \sqrt{ (TP )^{2}+ (PA)^{2}}$

Set TP = 30 and PA = 57 :

$TA = \sqrt{ 30 ^{2}+ 57.3^{2}}$

$= \sqrt{ 900 + 3,283}$

$= \sqrt{ 4,183}$

$\approx 65$

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Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Trapezoid

Figure NOT drawn to scale.

Refer to the above diagram, which shows Trapezoid TRAP with diagonal $\overline{TA}$ . To the nearest whole number, give the length of $\overline{TA}$ .

Possible Answers:

104

Correct answer:

Explanation:

To illustrate how to determine the correct length, draw a perpendicular segment from to $\overline{PA}$ , calling the point of intersection .

Trapezoid

$\overline{RX}$ divides the trapezoid into Rectangle TRXP and right triangle $\bigtriangleup RXA$ .

Opposite sides of a rectangle are congruent, so PX = TR = 40 .

$m \angle TRA = 135 ^{\circ }$ . The two angles of a trapezoid along the same leg - in particular, $\angle TRA$ and $\angle RAP$ - are supplementary, so

$m \angle RAP = 180 ^{\circ } - m \angle TRA$

$= 180 ^{\circ } - 135 ^{\circ }$

$= 45^{\circ }$

By the 45-45-45 Triangle Theorem,

XA = RX = TP =30

and

PA = PX + XA = 40+ 30 = 70

$\overline{TA}$ is the hypotenuse of right triangle $\bigtriangleup TPA$ , so by the Pythagorean Theorem, its length can be calculated to be

$TA = \sqrt{ (TP )^{2}+ (PA)^{2}}$

Set TP = 30 and PA = 70 :

$TA = \sqrt{ 30 ^{2}+ 70^{2}}$

$= \sqrt{ 900 +4,900}$

$= \sqrt{ 5,800}$

$\approx 76$

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Advanced Geometry : Trapezoids

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Example Question #3 : How To Find The Length Of The Diagonal Of A Trapezoid

Example Question #4 : How To Find The Length Of The Diagonal Of A Trapezoid

Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

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