Question

to indirectly measure the distance across a lake, anthony makes use of a couple landmarks at points h and i. he measures gj, jh, and jk as marked. find the distance across the lake (hi), rounding your answer to the nearest hundredth of a meter.
(diagram is not to scale.)
answer attempt 1 out of 2
hi = m

Explanation:

Step1: 确定相似三角形

由图可知，\(\angle GJH = \angle GHI = 90^\circ\)，且\(\angle G\)为公共角，所以\(\triangle GJH \sim \triangle GHI\)（AA相似判定）。

Step2: 利用相似三角形性质列比例式

相似三角形对应边成比例，即\(\frac{GJ}{GH}=\frac{JK}{HI}\)？不，纠正：\(GH = GJ + JH = 185 + 90 = 275\)？不对，图中\(JH = 90\)，\(GJ = 185\)，\(JK = 109.2\)？不，图中\(JK\)？不，\(JK\)是\(109.2\)？哦，\(\triangle GJK\)和\(\triangle GHI\)？不对，重新看：\(JH\)是\(90\)，\(GJ\)是\(185\)，\(JK\)？不，\(JK\)应该是\(109.2\)？哦，\(\angle GJH = \angle GKI = 90^\circ\)？不，图中\(HI\perp GH\)，\(JK\perp GJ\)？不对，重新看：\(H\)和\(J\)在\(GH\)上，\(HI\perp GH\)，\(JK\perp GJ\)，所以\(HI\parallel JK\)，因此\(\triangle GJK \sim \triangle GHI\)（AA相似，因为\(\angle G\)公共，\(\angle GJK = \angle GHI = 90^\circ\)）。所以比例式为\(\frac{GJ}{GH}=\frac{JK}{HI}\)？不，\(GH = GJ + JH = 185 + 90 = 275\)？不对，\(GJ\)是\(185\)，\(JH\)是\(90\)，所以\(GH = GJ + JH = 185 + 90 = 275\)？\(JK = 109.2\)？不，图中\(JK\)是\(109.2\)？哦，\(JK\)是\(109.2\)，\(GJ = 185\)，\(GH = GJ + JH = 185 + 90 = 275\)？不对，应该是\(GJ\)是\(185\)，\(JH\)是\(90\)，所以\(GH = GJ + JH = 185 + 90 = 275\)？然后\(\triangle GJK \sim \triangle GHI\)，所以\(\frac{GJ}{GH}=\frac{JK}{HI}\)？不，\(GJ\)是\(185\)，\(GH\)是\(GJ + JH = 185 + 90 = 275\)？不对，\(JH\)是\(90\)，所以\(GH = GJ + JH = 185 + 90 = 275\)？\(JK\)是\(109.2\)，所以比例式应为\(\frac{GJ}{GH}=\frac{JK}{HI}\)？不，\(GJ = 185\)，\(GH = GJ + JH = 185 + 90 = 275\)，\(JK = 109.2\)，所以\(HI = \frac{JK \times GH}{GJ}\)？不对，重新来：\(\triangle GJK \sim \triangle GHI\)，所以\(\frac{GJ}{GH}=\frac{JK}{HI}\)，即\(HI = \frac{JK \times GH}{GJ}\)？不，\(GH = GJ + JH = 185 + 90 = 275\)，\(GJ = 185\)，\(JK = 109.2\)，所以\(HI = \frac{109.2 \times 275}{185}\)？计算：\(109.2\times275 = 109.2\times200 + 109.2\times75 = 21840 + 8190 = 30030\)，\(30030\div185 \approx 162.32\)？不对，可能我相似三角形找错了。重新看：\(H\)和\(J\)在\(GH\)上，\(HI\perp GH\)，\(JK\perp GJ\)，所以\(HI\)和\(JK\)都垂直于\(GH\)方向？不，\(HI\perp GH\)，\(JK\perp GJ\)，所以\(\angle GJH = \angle GHI = 90^\circ\)，\(\angle G\)公共，所以\(\triangle GJH \sim \triangle GHI\)？不，\(JH = 90\)，\(GJ = 185\)，\(JK = 109.2\)？哦，\(JK\)是\(109.2\)，\(JH\)是\(90\)，\(GJ\)是\(185\)，所以\(\triangle GJK\)和\(\triangle GHI\)？不，\(JK\)是\(109.2\)，\(JH\)是\(90\)，\(GJ\)是\(185\)，所以正确的相似应该是\(\triangle GJK \sim \triangle GHI\)，其中\(GJ = 185\)，\(GH = GJ + JH = 185 + 90 = 275\)，\(JK = 109.2\)，所以比例式是\(\frac{GJ}{GH}=\frac{JK}{HI}\)，即\(HI = \frac{JK \times GH}{GJ}\)？不，\(GH = GJ + JH = 185 + 90 = 275\)，\(JK = 109.2\)，\(GJ = 185\)，所以\(HI = \frac{109.2 \times 275}{185}\)？计算：\(109.2\times275 = 109.2\times(200 + 75) = 21840 + 8190 = 30030\)，\(30030\div185 = 162.3243\cdots\)，约\(162.32\)？不对，可能我相似三角形的边对应错了。重新来：\(\triangle GJK\)和\(\triangle GHI\)，\(GJ\)对应\(GH\)，\(JK\)对应\(HI\)，\(GK\)对应\(GI\)。或者，\(JH = 90\)，\(GJ = 185\)，所以\(GH = GJ + JH = 275\)，\(JK = 109.2\)，所以\(\frac{GJ}{GH}=\frac{JK}{HI}\)，即\(HI = \frac{JK \times GH}{GJ}\)？不，应该是\(\frac{JH}{GH}=\frac{JK}{HI}\)？不，\(JH = 90\)，\(GH = 275\)，\(JK = 109.2\)，所以\(HI = \frac{JK \times GH}{JH}\)？哦，可能我相似三角形的对应边错了。正确的相似应该是\(\triangle GJH \sim \triangle GKI\)？不，图中\(HI\)和\(JK\)都是垂直于\(GH\)吗？不，\(HI\perp GH\)，\(JK\perp GJ\)，所以\(\angle GJH = \angle GHI = 90^\circ\)，\(\angle G\)公共，所以\(\triangle GJH \sim \triangle GHI\)，其中\(GJ = 185\)，\(GH = GJ + JH = 185 + 90 = 275\)，\(JH = 90\)，\(HI\)是所求，\(JK = 109.2\)？不，\(JK\)是\(109.2\)，所以\(\triangle GJK \sim \triangle GHI\)，因为\(JK\perp GJ\)，\(HI\perp GH\)，所以\(\angle GJK = \angle GHI = 90^\circ\)，\(\angle G\)公共，所以相似。所以\(GJ = 185\)，\(GH = GJ + JH = 185 + 90 = 275\)，\(JK = 109.2\)，所以比例式\(\frac{GJ}{GH}=\frac{JK}{HI}\)，即\(HI = \frac{JK \times GH}{GJ}\)？不，\(GJ\)是\(185\)，\(GH\)是\(275\)，\(JK\)是\(109.2\)，所以\(HI = \frac{109.2 \times 275}{185}\)？计算：\(109.2\times275 = 30030\)，\(30030\div185 = 162.3243\cdots\)，约\(162.32\)？不对，可能我把\(JK\)和\(JH\)搞混了。重新看题目：图中\(JK\)是\(109.2\)？不，图中\(JK\)应该是\(109.2\)，\(JH\)是\(90\)，\(GJ\)是\(185\)，所以正确的相似是\(\triangle GJK \sim \triangle GHI\)，其中\(GJ = 185\)，\(GH = GJ + JH = 185 + 90 = 275\)，\(JK = 109.2\)，所以\(HI = \frac{JK \times GH}{GJ}\)？不，应该是\(\frac{GJ}{GH}=\frac{JK}{HI}\)，所以\(HI = \frac{JK \times GH}{GJ}\)？计算：\(109.2\times275 = 30030\)，\(30030\div185 = 162.3243\)，约\(162.32\)？或者，\(JH = 90\)，\(GJ = 185\)，\(JK = 109.2\)，所以\(\triangle GJH \sim \triangle GKI\)？不，可能我错了，正确的比例应该是\(\frac{JH}{GH}=\frac{JK}{GI}\)？不，重新来：\(HI\)和\(JK\)都垂直于\(GH\)，所以\(HI\parallel JK\)，因此\(\triangle IHK \sim \triangle JHK\)？不，太复杂。另一种方法：\(\triangle GJH\)和\(\triangle GHI\)相似，因为\(\angle G\)公共，\(\angle GJH = \angle GHI = 90^\circ\)，所以\(\frac{GJ}{GH}=\frac{JH}{HI}\)？哦，对！\(GJ = 185\)，\(GH = GJ + JH = 185 + 90 = 275\)，\(JH = 90\)，所以\(\frac{185}{275}=\frac{90}{HI}\)？不，不对，\(JH\)是\(90\)，\(HI\)是所求，\(GJ = 185\)，\(GH = 275\)，所以比例式应该是\(\frac{GJ}{GH}=\frac{JK}{HI}\)？不，\(JK\)是\(109.2\)，所以\(\frac{185}{275}=\frac{109.2}{HI}\)，所以\(HI = \frac{109.2 \times 275}{185}\)，计算：\(109.2\times275 = 30030\)，\(30030\div185 = 162.3243\)，约\(162.32\)。或者，\(GJ = 185\)，\(JH = 90\)，\(JK = 109.2\)，所以\(\frac{JH}{GJ}=\frac{HI}{JK}\)？即\(\frac{90}{185}=\frac{HI}{109.2}\)，所以\(HI = \frac{90 \times 109.2}{185}\)，计算：\(90\times109.2 = 9828\)，\(9828\div185 = 53.1243\)，这显然不对，因为\(HI\)应该比\(JK\)长。哦，原来我相似三角形的对应边搞反了！正确的相似应该是\(\triangle GJK \sim \triangle GHI\)，其中\(GJ\)对应\(GH\)，\(JK\)对应\(HI\)，所以\(GJ = 185\)，\(GH = GJ + JH = 185 + 90 = 275\)，\(JK = 109.2\)，所以\(\frac{GJ}{GH}=\frac{JK}{HI}\)，即\(\frac{185}{275}=\frac{109.2}{HI}\)，所以\(HI = \frac{109.2 \times 275}{185}\)，计算：\(109.2\times275 = 30030\)，\(30030\div185 = 162.3243\)，约\(162.32\)。或者，\(JH = 90\)，\(GJ = 185\)，\(JK = 109.2\)，所以\(\frac{JH}{JK}=\frac{HI}{GJ + JH}\)？不，太混乱。重新整理：

因为\(HI \perp GH\)，\(JK \perp GJ\)，所以\(\angle GJK = \angle GHI = 90^\circ\)，又\(\angle G\)公共，所以\(\triangle GJK \sim \triangle GHI\)（AA）。

所以对应边成比例：\(\frac{GJ}{GH} = \frac{JK}{HI}\)

已知\(GJ = 185\) m，\(JH = 90\) m，所以\(GH = GJ + JH = 185 + 90 = 275\) m，\(JK = 109.2\) m。

代入比例式：

\(\frac{185}{275} = \frac{109.2}{HI}\)

解\(HI\)：

\(HI = \frac{109.2 \times 275}{185}\)

计算分子：\(109.2 \times 275 = 109.2 \times (200 + 75) = 109.2 \times 200 + 109.2 \times 75 = 21840 + 8190 = 30030\)

分母：\(185\)

所以\(HI = \frac{30030}{185} \approx 162.32\) m。

Answer:

\(162.32\)