Question

given △hpr is shown in figure. l is a midpoint of hp and hr || lt
which of the statements are incorrect?
a. △hpr ≅ △lpt d. \\(\frac{lp}{lh} = \frac{lt}{hr}\\)
b. m < t = m < r e. \\(\frac{pt}{lt} = \frac{pl}{lh}\\)
c. \\(\frac{lp}{lh} = \frac{pt}{tr}\\) f. \\(\frac{pl}{ph} = \frac{pt}{pr}\\)

Explanation:

Step1: Identify midpoint property

Since L is the midpoint of HP, $PL = LH$, so $\frac{LP}{LH}=1$.

Step2: Apply parallel line similarity

Because $HR \parallel LT$, $\triangle HPR \sim \triangle LPT$ by AA similarity (corresponding angles are equal).

Step3: Analyze each option

Option A:

$\triangle HPR \cong \triangle LPT$ is incorrect. Similar triangles are only congruent if corresponding sides are equal, but $LP = \frac{1}{2}HP$, so they are not congruent.

Option B:

$m\angle T = m\angle R$ is correct. Corresponding angles of similar triangles are equal.

Option C:

$\frac{LP}{LH} = \frac{PT}{TR}$. Since $LP=LH$, $\frac{LP}{LH}=1$. Also, T is the midpoint of PR (from midsegment theorem, as L is midpoint of HP and $HR\parallel LT$), so $PT=TR$, $\frac{PT}{TR}=1$. This is correct.

Option D:

$\frac{LP}{HR} = \frac{LT}{HR}$ is incorrect. $\frac{LP}{LH}=1$, and $LT = \frac{1}{2}HR$ (midsegment length is half the parallel side), so $\frac{LT}{HR}=\frac{1}{2}
eq 1$.

Option E:

$\frac{PT}{LT} = \frac{PL}{LH}$. $\frac{PL}{LH}=1$, but $\frac{PT}{LT}$ is not 1 (PT is a side of the small triangle, LT is the midsegment, they are not equal in length). This is incorrect.

Option F:

$\frac{PL}{PH} = \frac{PT}{PR}$. $\frac{PL}{PH}=\frac{1}{2}$ (L is midpoint), $\frac{PT}{PR}=\frac{1}{2}$ (T is midpoint). This is correct.

Answer:

A. $\Delta HPR \simeq \Delta LPT$, D. $\frac{LP}{LH} = \frac{LT}{HR}$, E. $\frac{PT}{LT} = \frac{PL}{LH}$