Math / Geometry

QuestionUse the diagram shown and the glven information to decide in each case whether $\overline{A D}$ is a perpendicular bisector, an angle bisector, a median, or an altitude of $\triangle A B C$ .
1. $\overline{D B} \cong \overline{D C}$ medicin
2. $\angle B A D \cong \angle C A D$ angle bisector
3. $\overline{D B} \cong \overline{D C}$ and $\overline{A D} \perp \overline{B C}$
4. $\overline{A D} \perp \overline{B C}$
5. $\triangle B A D \cong \triangle C A D$

Studdy Solution

STEP 1

1. We are given a triangle $\triangle ABC$ .
2. We need to determine the role of line segment $\overline{AD}$ in relation to $\triangle ABC$ based on given conditions.
3. Possible roles for $\overline{AD}$ include: perpendicular bisector, angle bisector, median, or altitude.

STEP 2

1. Analyze condition 1: $\overline{DB} \cong \overline{DC}$ .
2. Analyze condition 2: $\angle BAD \cong \angle CAD$ .
3. Analyze condition 3: $\overline{DB} \cong \overline{DC}$ and $\overline{AD} \perp \overline{BC}$ .
4. Analyze condition 4: $\overline{AD} \perp \overline{BC}$ .
5. Analyze condition 5: $\triangle BAD \cong \triangle CAD$ .

STEP 3

Condition 1 states $\overline{DB} \cong \overline{DC}$ . This implies that point $D$ is equidistant from points $B$ and $C$ . Therefore, $\overline{AD}$ is the perpendicular bisector of $\overline{BC}$ if it is also perpendicular to $\overline{BC}$ . Without the perpendicular condition, $\overline{AD}$ is a median.

STEP 4

Condition 2 states $\angle BAD \cong \angle CAD$ . This implies that $\overline{AD}$ divides $\angle BAC$ into two equal angles. Therefore, $\overline{AD}$ is an angle bisector of $\triangle ABC$ .

STEP 5

Condition 3 states $\overline{DB} \cong \overline{DC}$ and $\overline{AD} \perp \overline{BC}$ . Since $\overline{AD}$ is equidistant from $B$ and $C$ and also perpendicular to $\overline{BC}$ , $\overline{AD}$ is a perpendicular bisector of $\overline{BC}$ .

STEP 6

Condition 4 states $\overline{AD} \perp \overline{BC}$ . This implies that $\overline{AD}$ is an altitude of $\triangle ABC$ , as it is perpendicular to the base $\overline{BC}$ .

STEP 7

Condition 5 states $\triangle BAD \cong \triangle CAD$ . This implies that $\overline{AD}$ is both an angle bisector and a median, as it divides $\triangle ABC$ into two congruent triangles.
The roles of $\overline{AD}$ in each case are:
1. Median
2. Angle bisector
3. Perpendicular bisector
4. Altitude
5. Angle bisector and median

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Studdy Solution

STEP 1

1. We are given a triangle $\triangle ABC$ .
2. We need to determine the role of line segment $\overline{AD}$ in relation to $\triangle ABC$ based on given conditions.
3. Possible roles for $\overline{AD}$ include: perpendicular bisector, angle bisector, median, or altitude.

STEP 2

STEP 3

STEP 4

Condition 2 states $\angle BAD \cong \angle CAD$ . This implies that $\overline{AD}$ divides $\angle BAC$ into two equal angles. Therefore, $\overline{AD}$ is an angle bisector of $\triangle ABC$ .

STEP 5

Condition 3 states $\overline{DB} \cong \overline{DC}$ and $\overline{AD} \perp \overline{BC}$ . Since $\overline{AD}$ is equidistant from $B$ and $C$ and also perpendicular to $\overline{BC}$ , $\overline{AD}$ is a perpendicular bisector of $\overline{BC}$ .

STEP 6

Condition 4 states $\overline{AD} \perp \overline{BC}$ . This implies that $\overline{AD}$ is an altitude of $\triangle ABC$ , as it is perpendicular to the base $\overline{BC}$ .

STEP 7

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

Condition 2 states ∠BAD≅∠CAD \angle BAD \cong \angle CAD ∠BAD≅∠CAD. This implies that AD‾ \overline{AD} AD divides ∠BAC \angle BAC ∠BAC into two equal angles. Therefore, AD‾ \overline{AD} AD is an angle bisector of △ABC \triangle ABC △ABC.

STEP 5

STEP 6

Condition 4 states AD‾⊥BC‾ \overline{AD} \perp \overline{BC} AD⊥BC. This implies that AD‾ \overline{AD} AD is an altitude of △ABC \triangle ABC △ABC, as it is perpendicular to the base BC‾ \overline{BC} BC.

STEP 7

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Condition 2 states $\angle BAD \cong \angle CAD$ . This implies that $\overline{AD}$ divides $\angle BAC$ into two equal angles. Therefore, $\overline{AD}$ is an angle bisector of $\triangle ABC$ .

Condition 4 states $\overline{AD} \perp \overline{BC}$ . This implies that $\overline{AD}$ is an altitude of $\triangle ABC$ , as it is perpendicular to the base $\overline{BC}$ .