Question

in δxyz, ∠x=35° and ∠y=21°. ∠xwz=90° and xy=63. find the length of xw to the nearest 10th.

Explanation:

Step1: Find ∠Z in △XYZ

In a triangle, the sum of angles is \(180^\circ\). So, \(\angle Z = 180^\circ - \angle X - \angle Y = 180^\circ - 35^\circ - 21^\circ = 124^\circ\). But maybe we can use trigonometry with the right triangle. Let's denote \(XW = x\), \(WY = 63 - x\), and \(ZW = h\) (height).

In \(Rt\triangle XWZ\), \(\tan(35^\circ) = \frac{h}{x}\) ⇒ \(h = x\tan(35^\circ)\).

In \(Rt\triangle YWZ\), \(\tan(21^\circ) = \frac{h}{63 - x}\) ⇒ \(h = (63 - x)\tan(21^\circ)\).

Step2: Equate the two expressions for h

Since both equal \(h\), we have:
\(x\tan(35^\circ) = (63 - x)\tan(21^\circ)\)

Expand the right side: \(x\tan(35^\circ) = 63\tan(21^\circ) - x\tan(21^\circ)\)

Bring terms with \(x\) to left: \(x\tan(35^\circ) + x\tan(21^\circ) = 63\tan(21^\circ)\)

Factor \(x\): \(x(\tan(35^\circ) + \tan(21^\circ)) = 63\tan(21^\circ)\)

Step3: Solve for x

\(x = \frac{63\tan(21^\circ)}{\tan(35^\circ) + \tan(21^\circ)}\)

Calculate \(\tan(35^\circ) \approx 0.7002\), \(\tan(21^\circ) \approx 0.3839\)

Substitute values: \(x = \frac{63 \times 0.3839}{0.7002 + 0.3839} = \frac{24.1857}{1.0841} \approx 22.3\) (Wait, no, maybe I mixed up. Wait, actually, let's re-examine. Wait, maybe using the Law of Sines in △XYZ first.

Law of Sines: \(\frac{XZ}{\sin(21^\circ)} = \frac{XY}{\sin(124^\circ)}\)

\(XZ = \frac{63 \times \sin(21^\circ)}{\sin(124^\circ)}\)

\(\sin(21^\circ) \approx 0.3584\), \(\sin(124^\circ) = \sin(56^\circ) \approx 0.8290\)

\(XZ \approx \frac{63 \times 0.3584}{0.8290} \approx \frac{22.5792}{0.8290} \approx 27.24\)

Then in \(Rt\triangle XWZ\), \(\cos(35^\circ) = \frac{XW}{XZ}\) ⇒ \(XW = XZ\cos(35^\circ)\)

\(XW \approx 27.24 \times \cos(35^\circ)\), \(\cos(35^\circ) \approx 0.8192\)

\(XW \approx 27.24 \times 0.8192 \approx 22.3\)? Wait, no, maybe better to use the first method.

Wait, first method:

\(x = \frac{63 \times 0.3839}{0.7002 + 0.3839} = \frac{24.1857}{1.0841} \approx 22.3\)? Wait, no, wait:

Wait, \(\tan(35) ≈ 0.7002\), \(\tan(21) ≈ 0.3839\)

So denominator: \(0.7002 + 0.3839 = 1.0841\)

Numerator: \(63 * 0.3839 = 24.1857\)

24.1857 / 1.0841 ≈ 22.3? Wait, but let's check with Law of Sines again.

Law of Sines in △XYZ: \(\frac{XZ}{\sin(21^\circ)} = \frac{XY}{\sin(124^\circ)}\)

\(XZ = \frac{63 \times \sin(21^\circ)}{\sin(124^\circ)} = \frac{63 \times 0.3584}{0.8290} ≈ \frac{22.5792}{0.8290} ≈ 27.24\)

Then in right triangle XWZ, \(XW = XZ \cos(35^\circ) ≈ 27.24 * 0.8192 ≈ 22.3\). Wait, but maybe I made a mistake in the first method. Wait, no, let's do the first method again.

From \(x\tan35 = (63 - x)\tan21\)

\(x*0.7002 = (63 - x)*0.3839\)

\(0.7002x = 63*0.3839 - 0.3839x\)

\(0.7002x + 0.3839x = 24.1857\)

\(1.0841x = 24.1857\)

\(x = 24.1857 / 1.0841 ≈ 22.3\). Wait, but let's check with another approach.

Alternatively, let's use the formula for the length of the segment in a triangle with an altitude. The length of XW can be found by \(XW = \frac{XY \times \cos(\angle X) \times \cos(\angle Y)}{\cos(\angle X) + \cos(\angle Y)}\)? No, better to use the two right triangles.

Wait, another way: Let’s let \(XW = x\), then \(WY = 63 - x\). In △XWZ, \(ZW = x \tan 35\). In △YWZ, \(ZW = (63 - x) \tan 21\). So:

\(x \tan 35 = (63 - x) \tan 21\)

\(x (\tan 35 + \tan 21) = 63 \tan 21\)

\(x = \frac{63 \tan 21}{\tan 35 + \tan 21}\)

Calculating:

\(\tan 21 ≈ 0.38386\), \(\tan 35 ≈ 0.70021\)

So numerator: \(63 * 0.38386 ≈ 24.1832\)

Denominator: \(0.70021 + 0.38386 ≈ 1.08407\)

\(x ≈ 24.1832 / 1.08407 ≈ 22.3\)

Wait, but let's check with more precise values.

\(\tan(21^\circ) = \tan(21) ≈ 0.383864035\)

\(\tan(35^\circ) = \tan(35) ≈ 0.700207538\)

Numerator: 63 * 0.383864035 ≈ 63 * 0.383864 ≈ 24.183432

Denominator: 0.700207538 + 0.383864035 ≈ 1.084071573

24.183432 / 1.084071573 ≈ 22.3 (rounded to nearest tenth)

Wait, but maybe I made a mistake in the Law of Sines approach. Wait, in △XYZ, using Law of Sines:

\(\frac{XZ}{\sin 21} = \frac{XY}{\sin 124}\)

\(XZ = \frac{63 \sin 21}{\sin 124}\)

\(\sin 21 ≈ 0.35836795\), \(\sin 124 ≈ \sin(56) ≈ 0.82903757\)

So \(XZ ≈ \frac{63 * 0.35836795}{0.82903757} ≈ \frac{22.57718085}{0.82903757} ≈ 27.233\)

Then in △XWZ, which is right-angled at W, \(\cos(35^\circ) = \frac{XW}{XZ}\) ⇒ \(XW = XZ \cos(35^\circ)\)

\(\cos(35^\circ) ≈ 0.819152044\)

So \(XW ≈ 27.233 * 0.819152044 ≈ 22.3\)

Yes, so both methods give approximately 22.3.

Answer:

\(22.3\)