How to find the length of the diagonal of a trapezoid - Advanced Geometry

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Question

What is the length of the diagonals of trapezoid ? Assume the figure is an isoceles trapezoid.

Answer

To find the length of the diagonal, we need to use the Pythagorean Theorem. Therefore, we need to sketch the following triangle within trapezoid :

We know that the base of the triangle has length . By subtracting the top of the trapezoid from the bottom of the trapezoid, we get:

Dividing by two, we have the length of each additional side on the bottom of the trapezoid:

$\frac{6: m}{2} = 3: m$

Adding these two values together, we get .

The formula for the length of diagonal uses the Pythagoreon Theorem:

, where is the point between and representing the base of the triangle.

Plugging in our values, we get:

$AC = \sqrt{97}: m$

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Question

Find the length of the diagonals of this isosceles trapezoid, with $\textup{Height}=8$ .

Answer

To find the length of the diagonals, split the top side into 3 sections as shown below:

The two congruent sections plus 8 adds to 14. , so the two congruent sections add to 6. They must each be 3. This means that the top of the right triangle with the diagonal as a hypotenuse must be 11, since .

We can solve for the diagonal, now pictured, using Pythagorean Theorem:

take the square root of both sides

$\sqrt{185}=C$

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Question

Find the length of both diagonals of this quadrilateral.

Answer

All of the lengths with one mark have length 5, and all of the side lengths with two marks have length 4. With this knowledge, we can add side lengths together to find that one diagonal is the hypotenuse to this right triangle:

Using Pythagorean Theorem gives:

take the square root of each side

$\sqrt{97}= C$

Similarly, the other diagonal can be found with this right triangle:

Once again using Pythagorean Theorem gives an answer of $\sqrt{65}$

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Question

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

Answer

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

The length of the base of $\bigtriangleup CAD$ , $\overline{AD}$ has to be found because is the length of the base of $\bigtriangleup BAE$ .

.

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

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

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

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

$7 = (AD)^{2}$

$\sqrt{7} = AD$

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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Question

Find the length of the diagonal of the isosceles trapezoid given below.

Answer

In order to calculate the length of the diagonal, we first must assume that the height is perpendicular to both the top and bottom of the trapezoid.

Knowing this, we can draw in the diagonal as shown below and use the Pythagorean Theorem to solve for the diagonal.

$Using\ the\ pythagorean\ theorem\ we\ get: 12^2+20^2=diagonal^2$

$This\ simplifies\ to\ become\ 144+400=diagonal^2$

We now take the square root of both sides:

$\sqrt544=\sqrt diagonal^2$

$\sqrt16*\sqrt34=diagonal$

$4\sqrt34=diagonal$

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Question

Figure NOT drawn to scale.

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

Answer

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

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

Opposite sides of a rectangle are congruent, so .

$m \angle TRA = 150 ^{\circ }$ . 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 , and

$XA = 30 \sqrt{3}$

$\approx 30 \cdot 1.7321$

$\approx 30 \cdot 1.7321$

$\approx 52$

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

$\overline{TA}$ 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 and :

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

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

$= \sqrt{9,364 }$

$\approx 97$

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Question

Figure NOT drawn to scale.

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

Answer

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

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

Opposite sides of a rectangle are congruent, so .

$m \angle TRA = 120 ^{\circ }$ . 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 , 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 and :

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

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

$= \sqrt{ 4,183}$

$\approx 65$

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Question

Figure NOT drawn to scale.

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

Answer

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

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

Opposite sides of a rectangle are congruent, so .

$m \angle TRA = 135 ^{\circ }$ . 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,

and

$\overline{TA}$ 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 and :

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

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

$= \sqrt{ 5,800}$

$\approx 76$

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