Finding $\angle QCA$ from Altitude $AM$ of Triangle $ABC$

In summary, altitude in a triangle is a line segment that is drawn from a vertex perpendicular to the opposite side, forming a right angle with the side it intersects. To find the altitude, you can use the formula "altitude = (2 * area) / base". The altitude divides the triangle into two right triangles, with the angle opposite the altitude being the complementary angle to the angle at the vertex where the altitude is drawn. To find the angle QCA, you can use trigonometric functions depending on the given information. Even if the altitude is outside of the triangle, you can still find the angle QCA by using the Pythagorean theorem and the appropriate trigonometric function.
  • #1
anemone
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If $Q$ is a point on the altitude $AM$ of triangle $ABC$, and that $\angle QBA=20^{\circ}$, $\angle QBC=40^{\circ}$ and $\angle QCB=30^{\circ}$, find $\angle QCA$.
 
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  • #2
anemone said:
If $Q$ is a point on the altitude $AM$ of triangle $ABC$, and that $\angle QBA=20^{\circ}$, $\angle QBC=40^{\circ}$ and $\angle QCB=30^{\circ}$, find $\angle QCA$.
Point Q must be the orthocenter of
$\triangle ABC $
we have :$\angle QCA+30+40=90$
$\therefore \angle QCA=20^o$
 
  • #3
Albert said:
Point Q must be the orthocenter of
$\triangle ABC $
we have :$\angle QCA+30+40=90$
$\therefore \angle QCA=20^o$

Well done, Albert! Well done!(Yes) And thanks for participating!
 

FAQ: Finding $\angle QCA$ from Altitude $AM$ of Triangle $ABC$

What is the definition of altitude in a triangle?

The altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side, or to the line containing the opposite side. It forms a right angle with the side it intersects.

How do you find the altitude of a triangle?

To find the altitude of a triangle, you can use the formula "altitude = (2 * area) / base". This formula applies to all types of triangles, whether they are equilateral, isosceles, or scalene.

What is the relationship between altitude and angle in a triangle?

The altitude of a triangle divides the triangle into two right triangles. The angle opposite the altitude is the complementary angle to the angle at the vertex where the altitude is drawn. This means that the sum of these two angles is always 90 degrees.

How do you find the angle QCA given the altitude AM of triangle ABC?

To find the angle QCA, you can use the trigonometric functions sine, cosine, or tangent, depending on the given information. For example, if you know the length of the altitude and the adjacent side, you can use the cosine function to find the angle QCA.

Can you find the angle QCA if the altitude AM is outside of triangle ABC?

Yes, you can still find the angle QCA even if the altitude AM is outside of the triangle. You can use the Pythagorean theorem to find the length of the adjacent side, and then use the appropriate trigonometric function to find the angle QCA.

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