In $$ riangle P Q R, S $$ is the mid-point of $$ Q R $$ and $$ \angle P T R=90^{\circ} $$ (a) Name the median and the altitude. (b) Is $$ Q T=T R $$ ?

Question

In $$ 	riangle P Q R, S $$ is the mid-point of $$ Q R $$ and $$ \angle P T R=90^{\circ} $$ (a) Name the median and the altitude. (b) Is $$ Q T=T R $$ ?

UpStudy AI · Accepted Answer

To solve the problem, let's analyze the given information about triangle $$ PQR $$:

1. **Known Conditions:**
   - $$ S $$ is the midpoint of $$ QR $$.
   - $$ \angle PTR = 90^{\circ} $$.

### (a) Name the median and the altitude.

- **Median:** A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. In this case, since $$ S $$ is the midpoint of $$ QR $$, the segment $$ PS $$ is the median from vertex $$ P $$ to side $$ QR $$.

- **Altitude:** An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Since $$ \angle PTR = 90^{\circ} $$, the segment $$ PT $$ is the altitude from vertex $$ P $$ to side $$ QR $$.

### (b) Is $$ QT = TR $$?

To determine if $$ QT = TR $$, we can analyze the triangle further:

- Since $$ S $$ is the midpoint of $$ QR $$, we have $$ QS = SR $$.
- The altitude $$ PT $$ creates two right triangles, $$ PQT $$ and $$ PRT $$.
- In right triangle $$ PTR $$, since $$ PT $$ is perpendicular to $$ QR $$, we can use the properties of right triangles.

However, without additional information about the lengths of the sides or angles, we cannot definitively conclude whether $$ QT = TR $$.

In general, if $$ PT $$ is the altitude and $$ S $$ is the midpoint, then $$ QT $$ and $$ TR $$ will be equal if $$ T $$ lies on the line segment $$ QR $$ such that it divides $$ QR $$ into two equal segments.

Thus, the answer to part (b) depends on the specific location of point $$ T $$ along line segment $$ QR $$. If $$ T $$ is the foot of the altitude from $$ P $$ to $$ QR $$, then $$ QT $$ will equal $$ TR $$ because the altitude from a vertex in a right triangle to the hypotenuse bisects the hypotenuse.

In conclusion:
- **Median:** $$ PS $$
- **Altitude:** $$ PT $$
- **Is $$ QT = TR $$?** Yes, if $$ T $$ is the foot of the altitude from $$ P $$ to $$ QR $$.
(a) The median is $$ PS $$ and the altitude is $$ PT $$. (b) Yes, $$ QT = TR $$ if $$ T $$ is the foot of the altitude from $$ P $$ to $$ QR $$.

In is the mid-point of and
(a) Name the median and the altitude.
(b) Is ?

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Geometry Questions

In is the mid-point of and (a) Name the median and the altitude. (b) Is ?