Question

1. given $\triangle prs \cong \triangle cfh$, find the values of $x$, $y$, and $z$.

Explanation:

Step1: Analyze congruent triangles

Since \(\triangle PRS \cong \triangle CFH\), corresponding sides and angles are equal.
- For side \(PS\) and \(FH\)? Wait, no, \(PS\) is \(2x - 7\), and looking at \(\triangle CFH\), the hypotenuse is 39? Wait, no, maybe \(PS\) corresponds to \(CH\)? Wait, no, the right triangle \(\triangle CFH\) has legs 24 and let's see, angle at \(H\) is \((6z - 4)^\circ\), and right angle at \(F\). Wait, maybe \(PS\) corresponds to \(FH\)? No, maybe \(PS\) corresponds to \(CH\)? Wait, the user's notes have \(2x - 7 = 34\)? Wait, maybe I misread. Wait, the right triangle \(\triangle CFH\): leg \(CF = 24\), hypotenuse \(CH = 39\), so the other leg \(FH\) can be calculated by Pythagoras: \(FH=\sqrt{39^2 - 24^2}=\sqrt{(39 - 24)(39 + 24)}=\sqrt{15\times63}=\sqrt{945}=3\sqrt{105}\)? No, that doesn't match. Wait, the user's notes have \(2x - 7 = 34\), so \(2x=41\)? No, wait the user's note says \(x = 23\) from \(2x - 7 = 34\) (since \(2\times23 -7=46 -7=39\)? Wait, no, 2x -7=39? Wait, maybe \(PS\) corresponds to \(CH = 39\). So \(2x -7 = 39\), then \(2x=46\), \(x=23\). That matches the user's note.

Step2: Find y (assuming y is a side, maybe \(PR = CF = 24\), so y=24? The user's note has \(y = 1\)? No, maybe y is an angle. Wait, \(\triangle PRS \cong \triangle CFH\), so corresponding angles. \(\triangle CFH\) is right-angled at F, so \(\angle F = 90^\circ\), so maybe \(\angle R = 90^\circ\). Then angle at \(H\) is \((6z - 4)^\circ\), and in \(\triangle CFH\), angles sum to \(180^\circ\), so \(90^\circ + 24^\circ\) related? Wait, no, legs 24 and FH, hypotenuse 39. Wait, 24-32-39? No, 24-30-39? No, 24² + 30²=576 + 900=1476≠1521=39². Wait, 24² + 35²=576 + 1225=1801≠1521. Wait, 15² + 36²=225 + 1296=1521=39². Oh! Wait, 39²=1521, 24²=576, 36²=1296, 15²=225. Wait, maybe I made a mistake. Wait, 24-32-40? No. Wait, the user's note has \(6z - 4 = 100\)? Wait, \(\triangle CFH\): angles sum to 180, right angle at F (90°), so \(\angle C + \angle H = 90^\circ\). Wait, if \(\angle H = (6z - 4)^\circ\), and \(\angle C\) can be found by cosine: \(\cos C = \frac{CF}{CH}=\frac{24}{39}=\frac{8}{13}\), so \(\angle C \approx 52.02^\circ\), then \(\angle H \approx 37.98^\circ\). But the user's note has \(6z - 4 = 100\)? No, maybe the triangle is not right-angled? Wait, the diagram shows a right angle at F (the square), so \(\angle F = 90^\circ\). Then \(\angle C + \angle H = 90^\circ\). If \(\angle H = (6z - 4)^\circ\), and maybe \(\angle C = 100^\circ\)? No, that can't be. Wait, the user's note has \(6z - 4 = 100\), so \(6z=104\)? No, 6z -4=100 → 6z=104 → z=104/6≈17.33. But the user's note says z=23? Wait, no, the user's note has a scribble: \(6z -4 -100\)? Maybe typo. Wait, let's go back.

Wait, the problem is \(\triangle PRS \cong \triangle CFH\). So corresponding sides: \(PR = CF\), \(RS = FH\), \(PS = CH\). Corresponding angles: \(\angle P = \angle C\), \(\angle R = \angle F\), \(\angle S = \angle H\).

\(\angle F = 90^\circ\) (right angle), so \(\angle R = 90^\circ\).

\(PS = CH = 39\), so \(2x -7 = 39\) → \(2x = 46\) → \(x = 23\) (matches user's note).

\(PR = CF = 24\), so if y is \(PR\), then y=24? But user's note has \(y = 1\), maybe y is an angle. Wait, \(\angle S = \angle H = (6z -4)^\circ\). In \(\triangle PRS\), if \(\angle R = 90^\circ\), \(\angle P\) and \(\angle S\) sum to 90°. Wait, maybe \(\angle P\) is equal to \(\angle C\), and \(\angle C\) can be calculated from \(\triangle CFH\): \(\cos C = \frac{CF}{CH} = \frac{24}{39} = \frac{8}{13}\), so \(\angle C \approx 52.02^\circ\), then \(\angle S = \angle H = 90^\circ - 52.02^\circ \approx 37.98^\circ\). But the user's note has \(6z -4 = 100\), which would be \(z = 17.33\), not matching. Wait, maybe the triangle is not right-angled? The diagram shows a square at F, so it is right-angled. Maybe the user's note has a mistake, but following the calculation for x: \(2x -7 = 39\) (since \(CH = 39\) is the hypotenuse, and \(PS\) corresponds to \(CH\)), so \(x = 23\). For z: if \(\angle H = 100^\circ\) (but that can't be in a right triangle), maybe the triangle is not right-angled? Wait, the square at F indicates right angle. Maybe the user's problem has different lengths. Alternatively, maybe \(PS\) corresponds to \(FH\), and \(FH\) is 34? Then \(2x -7 = 34\) → \(2x = 41\) → \(x = 20.5\), but the user's note says \(x = 23\) from \(2x -7 = 34\) (2*23 -7=39, not 34). So there's confusion, but following the user's note:

From \(2x -7 = 34\) (even though 2*23 -7=39), solving \(2x -7 = 34\) → \(2x=41\) → \(x=20.5\), but the user's note says \(x=23\) from \(2x -7 = 39\) (2*23 -7=39). So likely \(PS = CH = 39\), so \(2x -7 = 39\) → \(x=23\).

For y: if \(PR = CF = 24\), then y=24 (assuming y is the length of \(PR\)).

For z: if \(\angle H = 100^\circ\), then \(6z -4 = 100\) → \(6z=104\) → \(z=17.33\), but the user's note says \(z=23\) from \(6z -4 = 100\)? No, 6*23 -4=134, not 100. So maybe the user's note has a typo, but following the calculation for x as per the note: \(2x -7 = 34\) → \(x=23\) (since 2*23 -7=39, maybe the intended equation was \(2x -7 = 39\), so \(x=23\)).

Answer:

\(x = 23\), assuming \(2x - 7 = 39\) (corresponding side to \(CH = 39\)), \(y = 24\) (corresponding side to \(CF = 24\)), and for z, if we take the user's note \(6z - 4 = 100\) (even though it's inconsistent with right triangle), \(z = \frac{104}{6} \approx 17.33\), but likely a mistake. However, based on the calculation for x from \(2x -7 = 39\), \(x = 23\).