Question

in the following triangle, find length bw. round to the nearest tenth.
bw =

Explanation:

Step1: Identify the geometric theorem

This is a right triangle with an altitude to the hypotenuse, so we can use the geometric mean theorem (or altitude-on-hypotenuse theorem), which states that the length of the altitude (BW) is the geometric mean of the lengths of the two segments it divides the hypotenuse into. Wait, actually, another form: In a right triangle, the length of a leg is the geometric mean of the hypotenuse and the adjacent segment. Wait, maybe better to use the formula for the altitude: If we have a right triangle with hypotenuse \( c \), and the two segments of the hypotenuse are \( a \) and \( b \), then the altitude \( h \) is \( h=\frac{ab}{c} \)? Wait, no, let's check the given values. Wait, maybe the triangle is similar. Let's assume the right triangle, with legs 13.8 and 23? Wait, no, the diagram shows a right triangle, with one leg 23, another segment 18.4, and BW is the altitude. Wait, maybe the formula is \( BW = \frac{13.8 \times 18.4}{23} \)? Wait, no, let's recall: In a right triangle, the altitude to the hypotenuse is equal to the product of the legs divided by the hypotenuse. Wait, maybe the legs are 13.8 and 18.4? No, the hypotenuse is 23? Wait, maybe the triangle has sides: let's see, the hypotenuse is 23, one segment of the hypotenuse is 18.4, and the other leg is 13.8. Wait, the geometric mean theorem: In a right triangle, the length of each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Wait, no, the altitude is the geometric mean of the two segments. Wait, maybe I got the labels wrong. Let's re-express: Let the right triangle be \( \triangle ABC \) with right angle at \( B \), and altitude \( BW \) to hypotenuse \( AC \), so \( AC = 23 \), \( AW = 18.4 \), \( WC = 23 - 18.4 = 4.6 \)? No, that doesn't make sense. Wait, maybe the given lengths are: one leg is 13.8, the hypotenuse segment is 18.4, and the hypotenuse is 23. Wait, the formula for the altitude \( h \) in a right triangle is \( h = \frac{leg_1 \times leg_2}{hypotenuse} \). Wait, maybe the two legs are 13.8 and 18.4, and hypotenuse is 23? Let's check: \( 13.8^2 + 18.4^2 = 190.44 + 338.56 = 529 \), and \( 23^2 = 529 \). Oh! So that's a right triangle with legs 13.8 and 18.4, hypotenuse 23. Then the altitude to the hypotenuse \( BW \) is \( \frac{13.8 \times 18.4}{23} \).

Step2: Calculate the product of the legs

First, calculate \( 13.8 \times 18.4 \). Let's compute that: \( 13.8 \times 18.4 = (14 - 0.2) \times 18.4 = 14 \times 18.4 - 0.2 \times 18.4 = 257.6 - 3.68 = 253.92 \).

Step3: Divide by the hypotenuse

Now, divide by 23: \( \frac{253.92}{23} = 11.04 \). Wait, but that's not right. Wait, no, wait: the altitude to the hypotenuse is \( \frac{leg1 \times leg2}{hypotenuse} \). Wait, leg1 is 13.8, leg2 is 18.4, hypotenuse is 23. So \( 13.8 \times 18.4 = 253.92 \), then \( 253.92 / 23 = 11.04 \). But wait, the problem says "BW" – maybe I mixed up the labels. Wait, maybe the legs are 23 and 13.8, and hypotenuse is 18.4? No, that can't be, since 23 > 18.4. Wait, no, the earlier calculation: \( 13.8^2 + 18.4^2 = 190.44 + 338.56 = 529 = 23^2 \), so that's correct. So the right triangle has legs 13.8 and 18.4, hypotenuse 23. Then the altitude to the hypotenuse is \( (13.8 \times 18.4)/23 \). Let's compute that: 13.8 * 18.4 = 253.92; 253.92 / 23 = 11.04. But wait, the problem says "BW" – maybe the altitude is 11.0? Wait, no, 11.04 rounded to the nearest tenth is 11.0? Wait, 11.04 to the nearest tenth is 11.0? Wait, 11.04: the tenths place is 0, hundredths is 4, so we round down, so 11.0? Wait, no, 11.04: the first decimal is 0, second is 4. Wait, no, 13.8 * 18.4 = 253.92; 253.92 / 23 = 11.04. So to the nearest tenth, that's 11.0? Wait, no, 11.04 is 11.0 when rounded to the nearest tenth? Wait, no, 11.04: the tenths digit is 0, the hundredths is 4, so we keep the tenths digit as 0, so 11.0. Wait, but maybe I made a mistake in the formula. Wait, another way: the altitude to the hypotenuse is also equal to the geometric mean of the two segments. Wait, the hypotenuse is 23, one segment is 18.4, the other is 23 - 18.4 = 4.6. Then the altitude should be \( \sqrt{18.4 \times 4.6} \). Let's compute that: 18.4 * 4.6 = 84.64; square root of 84.64 is 9.2. Wait, that's different. Oh! I see my mistake. The legs are not 13.8 and 18.4. Wait, the right triangle: let's label it correctly. Let the right triangle be \( \triangle BDV \) with right angle at \( B \), and \( BW \) is the altitude to hypotenuse \( DV \), which has length 23. Then \( BW \) is the altitude, \( W \) is on \( DV \), and \( DW = 18.4 \), \( WV = 23 - 18.4 = 4.6 \). Then by the geometric mean theorem, \( BW^2 = DW \times WV \), so \( BW = \sqrt{18.4 \times 4.6} \). Let's compute 18.4 * 4.6: 18 * 4.6 = 82.8, 0.4 * 4.6 = 1.84, so total 82.8 + 1.84 = 84.64. Then \( \sqrt{84.64} = 9.2 \). Ah! That's the correct formula. I confused the legs with the segments. So the altitude is the geometric mean of the two segments of the hypotenuse. So \( BW = \sqrt{18.4 \times 4.6} = \sqrt{84.64} = 9.2 \). Wait, but where does 13.8 come into play? Oh, maybe 13.8 is one of the legs. Let's check: in a right triangle, the leg squared is equal to the hypotenuse times the adjacent segment. So if one leg is \( BV = 13.8 \), then \( BV^2 = DV \times WV \)? Wait, no, \( BV^2 = DV \times WV \)? Wait, no, the geometric mean theorem: each leg is the geometric mean of the hypotenuse and the segment adjacent to that leg. So \( BV^2 = DV \times WV \), and \( BD^2 = DV \times DW \). Let's check \( BV^2 = 13.8^2 = 190.44 \). \( DV \times WV = 23 \times 4.6 = 105.8 \). No, that's not equal. Wait, \( BD^2 = 18.4^2 = 338.56 \), \( DV \times DW = 23 \times 18.4 = 423.2 \). No, that's not equal. Wait, I must have mislabeled the triangle. Let's start over. The problem shows a right triangle with right angle at the bottom left, hypotenuse at the top, length 23. One leg is 13.8 (vertical), another segment is 18.4 (horizontal? No, the diagram has a right angle, and BW is the altitude. Wait, maybe the triangle has sides: leg 1: 13.8, leg 2: 18.4, hypotenuse: 23 (since 13.8² + 18.4² = 190.44 + 338.56 = 529 = 23²). So that's a right triangle with legs 13.8 and 18.4, hypotenuse 23. Then the altitude to the hypotenuse is (13.8 * 18.4)/23. Let's compute that: 13.8 * 18.4 = 253.92; 253.92 / 23 = 11.04. But then why is there a 18.4 segment? Wait, maybe the 18.4 is one of the legs, and 13.8 is the other, and hypotenuse 23. Then the altitude is (13.8 * 18.4)/23 = 11.04, which rounds to 11.0? But earlier, when I did the geometric mean of the segments, I got 9.2, which is wrong because I misidentified the segments. Wait, no, in this case, the altitude is (leg1 * leg2)/hypotenuse. So leg1 = 13.8, leg2 = 18.4, hypotenuse = 23. So (13.8 * 18.4)/23 = 253.92 / 23 = 11.04. So to the nearest tenth, that's 11.0? Wait, 11.04 rounded to the nearest tenth is 11.0? Wait, 11.04: the tenths place is 0, hundredths is 4, so we round down, so 11.0. But wait, 11.04 is closer to 11.0 than 11.1? Wait, no, 11.04: the first decimal is 0, the second is 4, so we keep the tenths digit as 0, so 11.0. But let's check with the geometric mean of the segments. If the hypotenuse is 23, and the altitude is 11.04, then the two segments of the hypotenuse should be (leg1)² / hypotenuse and (leg2)² / hypotenuse. So (13.8)² / 23 = 190.44 / 23 = 8.28, and (18.4)² / 23 = 338.56 / 23 = 14.72. Then 8.28 + 14.72 = 23, which matches the hypotenuse. Then the altitude is the geometric mean of 8.28 and 14.72: √(8.28 * 14.72) ≈ √(121.9) ≈ 11.04, which matches. So my initial mistake was misidentifying the segments. So the correct formula is (leg1 * leg2)/hypotenuse. So with leg1 = 13.8, leg2 = 18.4, hypotenuse = 23, the altitude BW is (13.8 * 18.4)/23 = 11.04, which rounds to 11.0. Wait, but the problem says "round to the nearest tenth". 11.04 to the nearest tenth is 11.0? Wait, 11.04: the tenths digit is 0, the hundredths is 4, so we round down, so 11.0. Alternatively, maybe the legs are 23 and 13.8, and hypotenuse 18.4? No, that can't be, since 23 > 18.4. So the correct calculation is (13.8 * 18.4)/23 = 11.04, so BW ≈ 11.0.

Step1: Recall the formula for altitude in a right triangle

In a right triangle, the altitude (\( h \)) to the hypotenuse is given by \( h = \frac{\text{leg}_1 \times \text{leg}_2}{\text{hypotenuse}} \). Here, the legs are \( 13.8 \) and \( 18.4 \), and the hypotenuse is \( 23 \).

Step2: Substitute the values into the formula

Substitute \( \text{leg}_1 = 13.8 \), \( \text{leg}_2 = 18.4 \), and \( \text{hypotenuse} = 23 \) into the formula:
\( BW = \frac{13.8 \times 18.4}{23} \)

Step3: Calculate the numerator

First, compute the product of the legs:
\( 13.8 \times 18.4 = 253.92 \)

Step4: Divide by the hypotenuse

Divide the result by the hypotenuse:
\( BW = \frac{253.92}{23} = 11.04 \)

Step5: Round to the nearest tenth

Round \( 11.04 \) to the nearest tenth. The hundredths digit is \( 4 \), which is less than \( 5 \), so we round down:
\( BW \approx 11.0 \)

Answer:

\( 11.0 \)