How to find the length of the diagonal of a square - GRE Quantitative Reasoning

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Question

If a square has a side length of 4, how long is the diagonal of the square?

Answer

A diagonal divides a square into two 45-45-90 triangles, which have lengths adhering to the ratio of x: x: x√2. Therefore, 4√2 is the correct answer as the diagonal represents the hypotenuse of the triangle. the Pythagorean theorem can also be used: 42+ 42 = c2.

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Question

If a square has a side length of √2, how long is the diagonal of the square?

Answer

A diagonal divides a square into two 45-45-90 triangles, which have lengths adhering to the ratio of x: x: x√2. Therefore, 2 is the correct answer as the diagonal represents the hypotenuse of the triangle. the Pythagorean theorem can also be used: √22+ √22 = c2.

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Question

A square has a side of length 5. What is the length of its diagonal?

Answer

The diagonal separates the square into two 45-45-90 right triangles. The problem can be solved by using the Pythagorean Theorem, _a_2 + _b_2 = _c_2. It can also be solved by recognizing the 45-45-90 special triangles, which have side ratios of x : x : x√2.

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Question

A square with width of $6\ m$ is inscribed in a circle. What is the total area inside the circle?

Answer

We know that each side of the square is 6, so use the Pythagorean Theorem to solve for the diagonal of the square. The diagonal of the square is also the diameter of the circle.

$6^{2}+6^{2} = d^{2}$

$72= d^{2}$

$\dpi{100} \small \sqrt{72}=d$

$\dpi{100} \small \sqrt{72}=\sqrt{36\times 2}=6\sqrt{2}$

$\dpi{100} \small 6\sqrt{2}=d$

Therefore the radius must be $\dpi{100} \small r=\frac{6\sqrt{2}}{2}\rightarrow 3\sqrt{2}$

Now let's find the area inside the circle using the radius.

$\dpi{100} \small A=\pi r^{2}$

$\dpi{100} \small A=\pi\left (3\sqrt{2} \right )^{2}$

$\dpi{100} \small A=\pi\left (3\sqrt{2} \right )^{2}\rightarrow \pi\left ( 9\times 2 \right )\rightarrow 18\pi$

$\dpi{100} \small A=18\pi$ meters2

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Question

Quantity A:

The diagonal of a square with a side-length of .

Quantity B:

The side-length of a square with a diagonal of .

Answer

Quantity A: The diagonal of a square with a side-length of 7.

Quantity B: The side-length of a square with a diagonal of 14.

Both quantities can be determined, so can the relationship.

Quantity A:

To determine the diagonal of a square, it's important to remember that a diagonal directly bisects a square from corner to corner. In other words, it bisects the corners, creating two triangles with 45:45:90 proportions, with the diagonal serving as the hypotenuses . If you remember your special triangles, then the side-side-hypotenuse measurements have a ratio of $1:1:\sqrt{2}$ .

In Quantity A, the side-length is 7. Following the proportion:

$1(7):1(7): \sqrt{2}(7)$

The diagonal equals $7\sqrt{2}$ .

Quantity B:

We can use the same ratio to figure out quantity B by substituting x for the unknown side-length quantity, which looks like this:

$1(x):1(x):\sqrt{2}(x) = 14$

To find x in this ratio, just isolate x in the hypotenuse:

$\sqrt{2}x = 14$

Divide by $\sqrt{2}$

$x = \frac{14}{\sqrt{2}}$

Now, how does Quantity A and Quantity B match up?

$\frac{14}{\sqrt{2}} \approx9.89$

$7\sqrt{2}\approx9.89$

On the surface, it looks like the two quanties are equal. But how do we prove it? Well, we know that $\sqrt{2}\times \sqrt{2} = 2$ . Therefore, we know that:

$7\sqrt{2}\times\sqrt{2}=14$

Divide both sides by $\sqrt{2}$

$7\sqrt{2} = \frac{14}{\sqrt{2}}$

Therefore, both quantities are equal.

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