Question

calculator
alex has two different size bookends for his bigger books and smaller books.
what is the length of the diagonal distance, x, of his smaller bookend?
enter your answer, as a decimal, in the box.
x = \boxed{} in.
(there are two images of bookends: the bigger one has height 10 in. and diagonal 12 in., the smaller one has height 6 in. and we need to find diagonal x)

Explanation:

Step1: Find the base length of the bigger bookend's triangle

The bigger bookend's triangle has height 10 in and hypotenuse 12 in. Using the Pythagorean theorem \(a^2 + b^2 = c^2\), where \(c = 12\) and \(a = 10\), we find the base \(b\):
\(b = \sqrt{12^2 - 10^2} = \sqrt{144 - 100} = \sqrt{44} = 2\sqrt{11}\)? Wait, no—wait, actually, the two bookends are similar triangles (same shape, different size). So the ratio of height to hypotenuse (or height to base) should be equal. Wait, the bigger bookend: height 10, hypotenuse 12? Wait, no, maybe the base is the same? Wait, looking at the diagram, the base of both triangles is the same? Wait, no, the first triangle: height 10, hypotenuse 12? Wait, no, maybe the first triangle has height 10, hypotenuse 12, and the second has height 6, hypotenuse \(x\), and the base is the same. Wait, no, actually, the two triangles are similar, so the ratio of corresponding sides is equal.

Wait, let's re-express: The first triangle (bigger bookend) is a right triangle with height \(h_1 = 10\) in, hypotenuse \(c_1 = 12\) in? Wait, no, that can't be, because \(10^2 + b^2 = 12^2\) would mean \(b = \sqrt{144 - 100} = \sqrt{44} \approx 6.633\). But the second triangle has height \(h_2 = 6\) in, and same base \(b\), so hypotenuse \(x\) would satisfy \(6^2 + b^2 = x^2\). But since \(b^2 = 12^2 - 10^2 = 44\), then \(x^2 = 6^2 + 44 = 36 + 44 = 80\), so \(x = \sqrt{80} \approx 8.944\)? Wait, no, that doesn't make sense. Wait, maybe the first triangle is height 10, base (let's say) \(b\), hypotenuse 12? No, that would mean \(b = \sqrt{12^2 - 10^2} = \sqrt{44}\), but the second triangle has height 6, same base \(b\), so hypotenuse \(x = \sqrt{6^2 + b^2} = \sqrt{36 + 44} = \sqrt{80} \approx 8.94\). But wait, maybe the two triangles are similar, so the ratio of height to hypotenuse is equal. So \(\frac{10}{12} = \frac{6}{x}\)? Wait, no, that would be inverse. Wait, similar triangles: corresponding sides are proportional. So height of first / height of second = hypotenuse of first / hypotenuse of second. So \(\frac{10}{6} = \frac{12}{x}\)? No, that would be \(\frac{10}{6} = \frac{12}{x}\) → \(10x = 72\) → \(x = 7.2\). Wait, that makes more sense. Because if the triangles are similar, the ratio of height to hypotenuse is constant. Let's check: First triangle: height 10, hypotenuse 12. Second: height 6, hypotenuse \(x\). So \(\frac{10}{12} = \frac{6}{x}\)? Wait, no, that would be \(\frac{10}{6} = \frac{12}{x}\) (corresponding sides: height1/height2 = hypotenuse1/hypotenuse2). So \(10x = 6 \times 12\) → \(10x = 72\) → \(x = 7.2\). Let's verify with Pythagoras. First triangle: height 10, hypotenuse 12, so base \(b = \sqrt{12^2 - 10^2} = \sqrt{44} \approx 6.633\). Second triangle: height 6, base same \(b \approx 6.633\), so hypotenuse \(x = \sqrt{6^2 + 6.633^2} = \sqrt{36 + 44} = \sqrt{80} \approx 8.944\). Wait, that's a contradiction. So I must have misinterpreted the diagram.

Wait, maybe the first triangle has height 10, base (let's say) \(b\), and hypotenuse 12? No, that's not possible because 10 < 12, but \(10^2 + b^2 = 12^2\) → \(b = \sqrt{44}\). The second triangle has height 6, base \(b\) (same base), so hypotenuse \(x = \sqrt{6^2 + b^2} = \sqrt{36 + 44} = \sqrt{80} \approx 8.94\). But that seems off. Alternatively, maybe the two triangles are similar with height 10 and 6, and hypotenuse 12 and \(x\), so the ratio of height to hypotenuse is \(10/12 = 5/6\), so for the smaller triangle, height 6, hypotenuse \(x\) should satisfy \(6/x = 10/12\) → \(10x = 72\) → \(x = 7.2\). But why the discrepancy?

Wait, maybe the first triangle is height 10, hypotenuse 12, and the second is height 6, hypotenuse \(x\), and they are similar, so the ratio of height to hypotenuse is equal. So \(10/12 = 6/x\) → \(x = (6 \times 12)/10 = 72/10 = 7.2\). Let's check with Pythagoras: If \(x = 7.2\), then base \(b = \sqrt{7.2^2 - 6^2} = \sqrt{51.84 - 36} = \sqrt{15.84} \approx 3.979\). For the first triangle, base \(b = \sqrt{12^2 - 10^2} = \sqrt{44} \approx 6.633\), which is not equal. So that can't be.

Wait, maybe the first triangle has height 10, base (let's say) \(b\), and hypotenuse 12, and the second has height 6, base \(b\) (same base), so hypotenuse \(x\). Then \(10^2 + b^2 = 12^2\) → \(b^2 = 44\). Then \(6^2 + b^2 = x^2\) → \(36 + 44 = x^2\) → \(x^2 = 80\) → \(x = \sqrt{80} \approx 8.944\), which is approximately 8.94. But the problem says "as a decimal", so maybe 8.94? Wait, but let's check the ratio of heights: 10 and 6, ratio 5:3. Then hypotenuse should be 12*(3/5) = 7.2, but that contradicts. So which is correct?

Wait, looking at the diagram: The first bookend (bigger) has a vertical side of 10 in, hypotenuse 12 in. The second (smaller) has vertical side 6 in, hypotenuse \(x\). The horizontal base is the same for both? No, the base is the same? Wait, no, the base is the same length. So the two triangles are right triangles with the same base, different heights. So base \(b\) is common. Then for the first triangle: \(10^2 + b^2 = 12^2\) → \(b^2 = 144 - 100 = 44\). For the second triangle: \(6^2 + b^2 = x^2\) → \(36 + 44 = x^2\) → \(x^2 = 80\) → \(x = \sqrt{80} \approx 8.944\), which is approximately 8.94. But let's confirm: \(\sqrt{80} = 4\sqrt{5} \approx 8.944\), so 8.94 when rounded to two decimal places, or 8.9 if one. Wait, but maybe the problem is that the two triangles are similar, so the ratio of height to hypotenuse is equal. Wait, 10/12 = 5/6, so 6/x = 5/6 → x = 36/5 = 7.2. But why the base is different? Maybe the diagram shows that the base is the same, but that's not the case. Wait, maybe I misread the diagram: The first triangle has height 10, hypotenuse 12, and the second has height 6, and the base is the same as the first's base? No, that would mean different bases. Wait, maybe the two triangles are similar, so the ratio of height to base is equal. Let's find the base of the first triangle: \(b = \sqrt{12^2 - 10^2} = \sqrt{44} \approx 6.633\). Then the ratio of height to base is 10 / 6.633 ≈ 1.507. For the second triangle, height 6, so base would be 6 / 1.507 ≈ 3.979, and hypotenuse \(x = \sqrt{6^2 + 3.979^2} \approx \sqrt{36 + 15.83} \approx \sqrt{51.83} \approx 7.2\). Ah! So that's the key: the triangles are similar, so the ratio of height to base is the same. So first triangle: height 10, base \(b_1 = \sqrt{12^2 - 10^2} = \sqrt{44}\), ratio \(10 / \sqrt{44} \approx 1.507\). Second triangle: height 6, base \(b_2 = 6 / 1.507 \approx 3.979\), hypotenuse \(x = \sqrt{6^2 + b_2^2} \approx 7.2\). So the triangles are similar, so the ratio of corresponding sides (height, base, hypotenuse) is equal. So the ratio of height 10 to height 6 is 10/6 = 5/3. Therefore, the hypotenuse of the smaller triangle is 12*(3/5) = 7.2. That makes sense because similar triangles have proportional sides. So the key is recognizing similarity: same shape, so ratios of corresponding sides are equal.

So Step 1: Identify the two triangles are similar (same angle, right triangles, so AA similarity). Step 2: Set up the proportion of corresponding sides (height / hypotenuse) for both triangles. Step 3: Solve for \(x\).

Step1: Recognize similar triangles

The two bookend triangles are right triangles and similar (same shape, different size), so the ratio of height to hypotenuse is equal.

Step2: Set up the proportion

For the bigger bookend: height \(h_1 = 10\) in, hypotenuse \(c_1 = 12\) in.
For the smaller bookend: height \(h_2 = 6\) in, hypotenuse \(x\).
Proportion: \(\frac{h_1}{c_1} = \frac{h_2}{x}\)

Step3: Solve for \(x\)

Substitute values: \(\frac{10}{12} = \frac{6}{x}\)
Cross-multiply: \(10x = 12 \times 6\)
\(10x = 72\)
\(x = \frac{72}{10} = 7.2\)

Answer:

7.2